Recall that a directed union of R∗-algebras is an R∗-algebra. This together
with Zorn’s lemma enables us to consider maximal R∗-algebras: For every
C∗-algebra $A$ there is a maximal R∗-subalgebra $B\subset A$ (in the sense
that if $B_{0}\subset A$ is an R∗-subalgebra of $A$ with $B\subset B_{0}$ then
$B=B_{0}$).

######  Question 9.6.

Is it possible to find a concrete example of a maximal R∗-subalgebra of
$B(\ell_{2})$ (without resorting to Zorn’s lemma)?

Remark that if $A\subset B(H)$ is an R∗-subalgebra then
$A+F(H)+{\mathbb{C}}\operatorname{id}_{H}$ is also an R∗-algebra. Thus a
maximal R∗-subalgebra $A\subset B(H)$ needs to satisfy
$F(H)+{\mathbb{C}}\operatorname{id}_{H}\subset A$.

By the type decomposition theorem of von Neumann algebras and Dye’s theorem,
it is easy to see that a pair of von Neumann algebras have mutually
orthoisomorphic projection lattices if and only if the von Neumann algebras
are Jordan ∗-isomorphic. We do not have a succinct decomposition result for
R∗-algebras. That is why the following question can possibly have a negative
answer.

######  Question 9.7.

Let $A,B$ be two R∗-algebras. Assume that ${\mathcal{P}}(A)$ and
${\mathcal{P}}(B)$ are orthoisomorphic. Does it follow that $A$ is Jordan
∗-isomorphic to $B$?

It is also of interest whether the “orthoisomorphic” in this question can be
replaced with “lattice isomorphic”. Example 8.4 implies that this is not true
in the non-unital case. On the other hand, the author’s result [41, Corollary
4.3] (see also [2, Corollary 1.5] by Ayupov and Kudaybergenov) implies that a
pair of certain von Neumann algebras have mutually isomorphic projection
lattices (as lattices) if and only if the von Neumann algebras are Jordan
∗-isomorphic. If we restrict ourselves to the case $A$ and $B$ are
sequentially ultramatricial R∗-algebras, Theorem 7.5 states that $A$ and $B$
are ∗-isomorphic if ${\mathcal{P}}(A)$ is lattice isomorphic to
${\mathcal{P}}(B)$. This motivates a further question concerning the non-
ultramatricial case:

######  Question 9.8.

Let $A,B$ be R∗-algebras with countable Hamel dimension such that
${\mathcal{P}}(A)$ is lattice isomorphic to ${\mathcal{P}}(B)$. Does it follow
that $A$ is ∗-isomorphic to $B$?

Every statement about (generalized) Boolean algebras gives rise to a question
about R∗-algebras. Below we propose several questions in this direction.

Before that, we remark that there are several other attempts to give a
framework for the theory of noncommutative Boolean algebras or noncommutative
Boolean spaces. One natural way is to detect the class of commutative
C∗-algebras that arise as $C_{0}(K)$ for zero-dimensional (locally) compact
Hausdorff spaces $K$. This class coincides with that of commutative
C∗-algebras with real rank zero (i.e., every self-adjoint operator is in the
norm closure of the collection of self-adjoint operators with finite spectra).
Thus one may consider C∗-algebras with real rank zero as noncommutative
analogs of zero-dimensional locally compact Hausdorff spaces. See [8, Section
V.3] for more information about real and stable rank.

Another way of formulating noncommutative Boolean algebras is to consider a
certain class of inverse semigroups, which in connection with groupoids can be
related to the theory of C∗-algebras. See [39] for the details. Yet another
direction of research is the theory of quantum logic initiated by Birkhoff and
von Neumann [5], see [17]. The main target of this theory is an orthomodular
lattice (usually assumed to be $\sigma$-complete), which is an
orthocomplemented lattice ${\mathcal{P}}$ such that $p\vee(q\wedge
p^{\perp})=q$ for any $p,q\in{\mathcal{P}}$ with $p\leq q$. This class of
lattices includes the projection lattices of von Neumann algebras.

By Stone’s theorem, zero-dimensional locally compact Hausdorff spaces are in
one-to-one correspondence with generalized Boolean algebras. Recall that our
main discovery in this paper is that R∗-algebras form noncommutative analogs
of generalized Boolean algebras. For this reason, the author guesses that
those C∗-algebras obtained by completing R∗-algebras can be considered as
noncommutative analogs of zero-dimensional locally compact Hausdorff spaces.
(However, we need to be aware of the fact that the completion, e.g. $K(H)$,
can have many different dense R∗-subalgebras.) Now it is natural to ask:

######  Question 9.9.

Which C∗-algebra arises as the completion of an R∗-algebra?

By definition (see Example 3.5) this class contains all AF C∗-algebras. Recall
that the completion of an R∗-algebra has real rank zero, and stable rank one
if unital. Is it true that the completion of every R∗-algebra is nuclear? If
every R∗-algebra is ultramatricial, then a C∗-algebra is the completion of
some R∗-algebra if and only if it is AF. Blackadar once wrote in the first
edition [6, 7.1] of his book on K-theory of C∗-algebras the sentence “AF
algebras are in some sense the “zero-dimensional” C∗-algebras”, but he added
to this sentence in the second edition [7, 7.1] the phrase “although real rank
zero is now regarded as the most natural noncommutative analog of zero-
dimensionality”.

Let us summarize several facts on Boolean algebras. Although Theorem 7.1 is
very important in the theory of AF C∗-algebras, it gives no better
understanding of commutative AF C∗-algebras/generalized Boolean algebras. This
fact can be observed in the following proposition.

######  Proposition 9.10.

Let ${\mathcal{S}}\subset 2^{X}$ be a generalized field of sets. As in Section
4 we identify $R({\mathcal{S}})$ with the space of functions $\\{f\colon
X\to{\mathbb{C}}\mid\\#f(X)<\infty,\,\,f^{-1}(z)\in{\mathcal{S}}\text{ for all
}z\in{\mathbb{C}}\setminus\\{0\\}\\}$.

  * •

$R({\mathcal{S}})$ has countable Hamel dimension if and only if
$R({\mathcal{S}})$ is separable if and only if ${\mathcal{S}}$ has countable
cardinality.

  * •

$K_{0}(R({\mathcal{S}}))$ is isomorphic to $\\{f\in R({\mathcal{S}})\mid
f(X)\subset{\mathbb{Z}}\\}$, and $K_{0}(R({\mathcal{S}}))_{+}$ to $\\{f\in
K_{0}(R({\mathcal{S}}))\mid f\geq 0\\}$.

  * •

$d(R({\mathcal{S}}))$ is the same as ${\mathcal{S}}$ as a set, and for $A,B\in
d(R({\mathcal{S}}))={\mathcal{S}}$, the sum $A+B$ is defined if and only if
$A\wedge B=0$, in which case we have $A+B=A\vee B$.

We omit the proof because it is easy and these facts should be well-known.
There exist many wild examples of Boolean algebras and Boolean spaces, which
implies the existence of wild commutative R∗-algebras.

######  Example 9.11.

In [38], as a byproduct of his structural theorem for countable Boolean
algebras, Ketonen proved that there exists a countable Boolean algebra
${\mathcal{S}}$ such that ${\mathcal{S}}$ is lattice isomorphic to
${\mathcal{S}}\times{\mathcal{S}}\times{\mathcal{S}}$ but not to
${\mathcal{S}}\times{\mathcal{S}}$. This is known as a counterexample to
Tarski’s cube problem. It follows that the commutative unital R∗-algebra
$R({\mathcal{S}})$ has countable Hamel dimension, and $R({\mathcal{S}})$ is
∗-isomorphic to $R({\mathcal{S}})\oplus R({\mathcal{S}})\oplus
R({\mathcal{S}})$ but not to $R({\mathcal{S}})\oplus R({\mathcal{S}})$.

######  Example 9.12.

It is known that every metrizable Boolean space $X$ that is homeomorphic to
$X\times X\times X$ is homeomorphic to $X\times X$. It is also known that
there exists a separable Boolean space $X$ that is homeomorphic to $X\times
X\times X$ but not to $X\times X$. See Trnková’s [50, 2.5]. Recall that for a
locally compact Hausdorff space $K$, $K$ is metrizable if and only if
$C_{0}(K)$ is separable. It follows that there exists no countable Boolean
algebra ${\mathcal{S}}$ such that $R({\mathcal{S}})$ is ∗-isomorphic to
$R({\mathcal{S}})\otimes R({\mathcal{S}})\otimes R({\mathcal{S}})$ but not to
$R({\mathcal{S}})\otimes R({\mathcal{S}})$, and there does exist such an
example if we drop the assumption of countability.

Are there similar examples in the noncommutative case?

######  Question 9.13.

Is there a simple or countable-dimensional R∗-algebra $A$ that is ∗-isomorphic
to $A\otimes A\otimes A$, but not to $A\otimes A$?

Question 9.3 might force us to consider only the case of ultramatricial
R∗-algebras. See also the Scottish Book [46] Problem 77.

Let us also consider the following similar question: Is there an R∗-algebra
$A$ that is ∗-isomorphic to ${\mathbb{M}}_{3}(A)$, but not to
${\mathbb{M}}_{2}(A)$? If we restrict ourselves to the simple purely atomic
case, this reduces to the following one. Is there an inner product space $V$
such that $V$ is isometrically isomorphic to the $\ell_{2}$-direct sum
$V\oplus V\oplus V$ but not to $V\oplus V$? This question is reminiscent of an
example by Gowers and Maurey in [28] of a Banach space $X$ that is isomorphic
to $X\oplus X\oplus X$ but not to $X\oplus X$ (as Banach spaces). In the
general setting of R∗-algebras, the infinite tensor product $\bigotimes_{n\geq
1}{\mathbb{M}}_{3}$ is an example that satisfies the condition of the
question. (The author thanks Masaki Izumi for pointing out this fact.)

In [34, Theorem 4.1] Kalton and Roberts proved that there exists a constant
$K>0$ such that for every Boolean algebra ${\mathcal{S}}$ and every map
$f\colon{\mathcal{S}}\to{\mathbb{R}}$ with $f(0)=0$ and $\lvert f(p\vee
q)-f(p)-f(q)\rvert\leq 1$ for any mutually orthogonal pair
$p,q\in{\mathcal{S}}$, there exists a finitely additive (signed) measure
$\mu\colon{\mathcal{S}}\to{\mathbb{R}}$ such that $\lvert
f(p)-\mu(p)\rvert\leq K$ for every $p\in{\mathcal{S}}$. This theorem was
applied in [34] to study a certain property of (quasi-)Banach spaces. By
comparing this with Theorem 8.9, it is fairly natural to formulate its
noncommutative version.

######  Question 9.14 (Noncommutative analog of the Kalton–Roberts theorem).

Is there a constant $K>0$ with the following property? For every unital
R∗-algebra $A$ and every map $f\colon{\mathcal{P}}(A)\to{\mathbb{R}}$ with
$f(0)=0$ and $\lvert f(p\vee q)-f(p)-f(q)\rvert\leq 1$ for any mutually
orthogonal pair $p,q\in{\mathcal{P}}(A)$, there exists a finitely additive
(signed) measure $\mu\colon{\mathcal{P}}(A)\to{\mathbb{R}}$ such that $\lvert
f(p)-\mu(p)\rvert\leq K$ for every $p\in{\mathcal{P}}(A)$.

It might be interesting to consider the same question for the projection
lattice of a von Neumann algebra.

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# Pattern selection and restricted vortex dynamics by spatial periodic forcing
in rapidly rotating Rayleigh-Bénard convection

Shan-Shan Ding  Hong-Lin Zhang  Dong-Tian Chen  Hao-Han Sun  Jin-Qiang Zhong
<EMAIL_ADDRESS>School of Physics Science and Engineering, Tongji
University, Shanghai 200092, China

(August 28, 2024)

###### Abstract

Pattern forming with externally imposed symmetry is ubiquitous in nature but
lightly studied. We present experimental studies of pattern formation and
selection by spatial periodic forcing in rapidly rotating convection. We
observe symmetric convection patterns in form of regular vortex lattice near
the instability onset, when the periodicity of the external forcing is set
close to the intrinsic vortex spacing. We show that the new patterns arise as
a dynamical process of imperfect bifurcation which can be well described by a
Ginzburg-Landau-like model. With increasing buoyancy strength the effect of
external forcing weakens, and the convective vortices evolve from a stationary
state to exhibit restricted and finally stochastic motions.

Pattern formatting phenomena are omnipresent in a wide variety of physical,
chemical, and biological systems Cross and Hohenberg (1993); Gollub and Langer
(1999); Zaikin and Zhabotinskii (1970); Field and Burger (1985); Murray
(1989). Nonequilibrium spatiotemporal patterns often arise through symmetry-
breaking bifurcations when an initially uniform system is driven internally
away from thermodynamic equilibrium. Many natural systems are, however, often
constrained with nonuniform boundaries having broken symmetry that may modify
the pattern formatting process and influence the intrinsic flow organization.
Examples include atmospheric convection rolls forming over mesoscale
topography Tian and Parker (2003), formation of Taylor columns over seamounts
that determine the overlying pattern of ice-cover in high-latitude oceans
Martin and Drucker (1997). Exploring the interaction between externally
imposed symmetries and intrinsic symmetries preferred by the system may shed
new light onto the complexity in pattern formatting process, and enable us to
induce, control or eliminate patterns using external forcing in various
systems Kelly and Pal (1978); Lowe et al. (1983); Coullet (1986); Dolnik et
al. (2001); Ismagilov et al. (2001); McCoy (2007); Seiden et al. (2008).

The fundamental physics of pattern formation have been studied over the past
few decades in carefully controlled experimental systems, such as in rotating
Rayleigh-Bénard convection (RBC) Cross and Hohenberg (1993); Zhong et al.
(1991); Bajaj et al. (1998), i. e., a fluid layer heated from below and
rotated about a vertical axis with angular velocity $\Omega_{\mathrm{D}}$.
When the temperature difference ${\Delta}T$ exceeds the onset
${\Delta}T_{\mathrm{c}}$($\Omega_{\mathrm{D}}$), convection pattern appears
first as straight rolls under slow rotations Kuppers and Lortz (1969); Hu et
al. (1997); Bajaj et al. (2002), which becomes unstable to the Kuppers-Lortz
instability when $\Omega_{\mathrm{D}}$ increases Hu et al. (1998); Sánchez-
Álvarez et al. (2005). Square (or hexagonal) patterns may form when the
dimensionless rotation rate, $\Omega{=}\Omega_{\mathrm{D}}H^{2}/\nu$ ($\nu$ is
the kinematic viscosity), reaches the order of 100 Bajaj et al. (1998). With
further increase in $\Omega$, the flow pattern becomes disordered Bajaj et al.
(2002). It is observed that under large rotation rates ($\Omega{\geq}10^{4}$)
the flow field near onset is characterized by columnar vortices that exhibit
Brownian-type random motion Noto et al. (2019); Chong et al. (2020); Ding et
al. (2021). Clearly in rapidly rotating convection the spatiotemporal
periodicity of the flow pattern is lost even close to onset. Previous studies
have shown that when flow patterns are modulated by a spatially periodic
perturbation, a commensurate state often arises in which the structural
periodicity of the flow rationally accommodates to that of the perturbation,
as a result of competition between the imposed and intrinsic patterns Lowe et
al. (1983); Ismagilov et al. (2001); Seiden et al. (2008). It raises naturally
the intriguing question whether flow patterns with spatial periodicity and
symmetry can be restored in rapidly rotating convection when external forcing
is applied.

In this Letter we report experimental observation of orderly flow patterns
forming under the control of externally imposed topographic forcing in rapidly
rotating convection. We use a convection apparatus that was designed for high-
resolution flow structure measurements in rotating RBC Chong et al. (2020);
Ding et al. (2021); Shi et al. (2020). We use a cylindrical cell that had a
diameter $d{=}235.0$ mm and a height $H{=}63.0$ mm, yielding an aspect ratio
$\Gamma{=}d/H{=}3.7$. The bottom plate of the sample, made of oxygen-free
copper, is finely machined to construct an array of thin cylinders extending
out from the plate surface. These raised cylinders are periodically spaced to
form a square- or hexagon-patterned surface. Flow patterns at a fluid depth of
$z{=}H/4$ are measured using the technique of particle image velocimetry (see
Supplemental Material SM  for detailed experimental methods). The experiment
is conducted with a constant Prandtl number Pr${=}\nu/{\kappa}{=}4.38$ and in
the range $2.0{\times}10^{6}{\leq}$Ra${\leq}1.0{\times}10^{8}$ of the Rayleigh
number $\mathrm{Ra}{=}{\alpha}g{\Delta}TH^{3}/{\kappa}{\nu}$. Here $g$ is the
gravitational acceleration, $\alpha$, $\kappa$ are respectively the isobaric
thermal expansion coefficient, thermal diffusivity of the fluid. Rotating
angular velocity of $0.6{\leq}{\Omega_{\mathrm{D}}}{\leq}2.0$ rad/s are used
thus $3.6{\times}10^{3}{\leq}{\Omega}{\leq}1.2{\times}10^{4}$. The reduced
Rayleigh number,
$\varepsilon{=}(\mathrm{Ra}{-}\mathrm{Ra}_{\mathrm{c}})/\mathrm{Ra}_{\mathrm{c}}$,
spans the range ${-}0.6{\leq}{\varepsilon}{\leq}13.5$. Here
$\mathrm{Ra_{c}}{=}\mathrm{Ra}({\Delta}T_{\mathrm{c}})$ is the onset of
convection Ons . The Froude number,
$\mathrm{Fr}{=}{\Omega}_{\mathrm{D}}^{2}d/2g$, covers the range
$4.4{\times}10^{{-}3}{\leq}\mathrm{Fr}{\leq}0.05$.

Figure 1: (a-c) Instantaneous vertical vorticity distribution
$\omega/\omega_{std}$. $\omega_{std}$ is the standard deviation of $\omega$.
Results are for the reference cell (a), the square-patterned (b) and the
hexagon-patterned (c) cells. $\Omega{=}1.12{\times}10^{4}$ and
$\varepsilon{=}0.49$. The distances between adjacent cylinders are
$\lambda{=}14.14$ mm (b) and $17.32$ mm (c). (d-f) Fourier spectra
$F(\vec{k})$ of the vorticity field, determined by $\omega(\vec{r})$ in the
central region of $60{\times}60$ mm2 shown in (a-c) respectively. The arrow in
(d) shows the mean radius $k_{0}$ of the crater-like structure. The arrows in
(e) and (f) represent the characteristic wave vectors
$\vec{k}_{\mathrm{f}}{=}\vec{k}_{i}(i{=}1,2...)$ of the imposed textures.
Movies for (a-c) are available SM .

Figure 1 presents the flow patterns at the measured fluid height with
$\Omega{=}1.12{\times}10^{4}$ and $\varepsilon{=}0.49$. When a flat bottom
plate is used (i. e., the reference cell without external forcing), the flow
fields of the vertical vorticity $\omega(\vec{r})$ are characterized by
columnar vortices, which exhibit stochastic horizontal motions as reported in
previous studies Noto et al. (2019); Chong et al. (2020); Ding et al. (2021).
Despite their random motion, the vortices maintain approximately a constant
distance $\lambda_{0}{\approx}13.15$ mm with their neighboring vortices (Fig.
1a). In the spatial Fourier spectrum $F(\vec{k})$ of vorticity field
calculated in central region (Fig. 1d), a crater-like structure with a radius
$k_{0}{=}1/\lambda_{0}$ is apparent in the spectrum $F(\vec{k})$, indicating
that the vortices are distributed with random orientations but with a
preferred spacing. When periodic topographic structures are constructed on the
bottom plate, they modulate both the local temperature and the shearing
interaction of the fluid with the solid surface, leading to new convection
patterns. Figures 1b and 1c show the vorticity fields when the bottom surface
is textured with a square and a hexagonal array of cylinders, respectively,
with their spacing $\lambda$ chosen close to the intrinsic wavelength of the
vorticity field $\lambda_{0}$. Since the fluid overlying the raised cylinders
is relatively hotter, upwelling vortices (i.e., cyclones when observed in the
lower half fluid layer Not ) tend to form above the cylinders, forming a 1:1
commensurate structure with respect to the bottom texture. The downwelling
vortices (anticyclones), however, appear in between the raised cylinders. In
the square-patterned cell (Fig. 1b), both the cyclones and anticyclones
constitute a regular square lattice. The flow pattern induced by a hexagonal
array of cylinders (Fig. 1c), however, consists of a hexagonal lattice of
anticyclones with a cyclone located at the hexagon center. Such patterns are
stationary and persist during the experiment. Figures 1e and 1f present the
Fourier spectra of the vorticity fields in the central region of Figs. 1b and
1c, respectively. In these spectra we see clear peaks located precisely at the
wave vectors $\vec{k}_{\mathrm{f}}{=}\vec{k}_{i}(i{=}1,2...)$ of the
periodically imposed textures. These peaks of $F(\vec{k})$ are all very sharp
and their amplitudes are approximately equal, suggesting that regular patterns
with prescribed periodicity and symmetry are developed. Near the sidewall
region ($r{\geq}100$ mm) where the imposed texture is absent, the flow field
is time-varying and the vortex dynamics therein is largely influenced by the
retrogradely traveling boundary zonal flow de Wit et al. (2020); Zhang et al.
(2020).

Figure 2: (a, b) Radial distribution function $g(r)$ of cyclones in the cell
textured with a square array of cylinders. Results are for various
$\varepsilon$ with a constant cylinder spacing $\lambda{=}14.14$ mm in (a),
and for various $\lambda$ with $\varepsilon{=}0.49$ in (b). The red, blue and
green arrows in (b) denote the main and subharmonic wavelengths $r_{ij}$ of
the imposed pattern for $\lambda{=}14.14,20.00$ and $28.28$ mm, respectively.
Dotted line in (b): result of the reference cell. (c) Contour plot of cross-
correlation coefficient $C$ of the vorticity field and bottom texture in the
$k_{f}{-}\varepsilon$ phase diagram. Open symbols represent the data points
measured in the square-patterned cell; color contours are estimated from
interpolation between these points. The horizontal dashed lines denote the
onset $\varepsilon{=}0$. Results are for $\Omega{=}1.12{\times}10^{4}$.

The observed spatial pattern can be quantified by the radial distribution
function $g(r)$ of the vortices, which is defined as the ratio of the actual
number of cyclones lying within an annulus region of $r$ and $r{+}{\delta}r$,
to the expected number for uniform vortex distribution Chong et al. (2020).
Figure 2a shows $g(r)$ for cyclonic distribution for various $\varepsilon$ in
the square-patterned cell with $\lambda{=}14.14$ mm. Near onset
($\varepsilon{=}0.09$) multiple sharp peaks appear in $g(r)$, which locate at
distances that match with the main and subharmonic wavelengths $r_{ij}$ of the
forced square pattern at the bottom plate, fulfilling the condition:
$r_{ij}{=}{\lambda}\sqrt{i^{2}{+}j^{2}}$, $(i,j{=}0,1,2...$ and
$i{+}j{\geq}1)$. With increasing $\varepsilon$, the peak amplitudes in $g(r)$
decrease while the peak widths increase, signifying a less regular flow
pattern. The multiple-peak structure is eventually flattened and $g(r)$
becomes close to a uniform distribution for $\varepsilon{\geq}4.0$, where we
see the vortices exhibit apparent horizontal motions. We examine as well the
role of the periodicity of external forcing on the convection pattern. Figure
2b presents results of $g(r)$ near onset ($\varepsilon{=}0.49$) for various
cylinder spacing $\lambda{=}10.00,14.14,20.00,28.28$ mm. We see multiple peaks
still appear at the main and subharmonic wavelengths $r_{ij}(\lambda)$. For
$\lambda{\geq}20.00$mm, the first peak is found near $r{=}\lambda_{0}$ which
is associated with the intrinsic wavelength of the flow field. When $\lambda$
is far apart from $\lambda_{0}$, the maxima of $g(r)$ become less dominant and
$g(r)$ approaches the result of the reference cell.

The degree of matching between the flow pattern and the bottom texture can be
evaluated through the cross-correlation coefficient $C$ of the vorticity field
and the bottom texture, defined as
$C{=}{\langle}\Delta{\omega}(\vec{r}){\cdot}\Delta{M}(\vec{r}){\rangle}/\sqrt{({\langle}\Delta{\omega}(\vec{r})^{2}\Delta{M}(\vec{r})^{2}{\rangle})}$,
where
$\Delta{\omega}(\vec{r}){=}{\omega}(\vec{r}){-}{\langle}{\omega}{\rangle}$,
$\Delta{M}(\vec{r}){=}M(\vec{r}){-}{\langle}M{\rangle}$, and $M(\vec{r}){=}0$
(or ${-}1$) for the flat (or raised) area, representing the bottom surface
profile. ${\langle}{\cdot}{\rangle}$ denotes a spatial average. Figure 2c
summarizes the results of $C$ in the square-patterned cell for varying
$\varepsilon$ and wave vector
$k_{\mathrm{f}}{=}{|}\vec{k}_{1}{|}{=}{|}\vec{k}_{2}{|}$ of the external
forcing. In this phase diagram, we see that $C(k_{\mathrm{f}},\varepsilon)$
has a single maximum ($C{\approx}0.7$) occurring at
($k_{\mathrm{m}}{=}70.7\mathrm{m}^{-1},\varepsilon_{\mathrm{m}}{=}0.09$),
which implies the optimal conditions for pattern selection. In the vicinity of
$(k_{\mathrm{m}},\varepsilon_{\mathrm{m}})$, the spatial distribution of the
vortices severely conforms to the bottom texture. The coefficient $C$
decreases if the control parameters $(k_{\mathrm{f}},\varepsilon)$ deviate
from $(k_{\mathrm{m}},\varepsilon_{\mathrm{m}})$. The decreasing of $C$ with
increasing $\varepsilon$ is slowest when a near-resonant external forcing
$(k_{\mathrm{f}}{=}k_{m}{\approx}k_{0})$ is chosen. The convection pattern and
the imposed texture becomes essentially uncorrelated (with $C{\leq}0.1$) when
$(k_{\mathrm{f}},\varepsilon)$ are set apart from
$(k_{\mathrm{m}},\varepsilon_{\mathrm{m}})$. Interestingly, when
$k_{\mathrm{f}}{\approx}k_{\mathrm{m}}$, $C$ remains well above zero for
$\varepsilon{<}0$, suggesting that under external forcing convection sets in
with finite amplitude in the subcritical regime.

We measure the time-averaged vorticity modulus
${\langle}{|}{\omega}{|}{\rangle}$ in an area of $65.8{\times}54.8$ mm2 at the
center of the cell, while slowly scanning ${\Delta}T$ in the range
$-0.4{\leq}\varepsilon{\leq}0.4$. Results of
${\langle}{|}{\omega}{|}{\rangle}(\varepsilon)$ for the reference cell and two
forced cells are shown in Fig. 3. Overall, these data suggest two distinct
types of bifurcations when $\varepsilon$ increases and crosses zero. The
reference cell data reveal a sharp transition from a non-convection state with
${\langle}{|}{\omega}{|}{\rangle}{=}0$ for $\varepsilon{\leq}0$, to a
convection state in which ${\langle}{|}{\omega}{|}{\rangle}$ increases rapidly
for $\varepsilon{>}0$. For both forced cells, however,
${\langle}{|}{\omega}{|}{\rangle}$ remains positive for
$\varepsilon{\geq}{-}0.4$ and grows relatively slowly with increasing
$\varepsilon$, suggesting a smooth transition. The three inset panels present
the vorticity fields captured in the three cells with approximately the same
subcriticality ($\varepsilon{\approx}{-}0.1$). They demonstrate that while the
fluid is still quiescent in the reference cell, apparent square (hexagonal)
lattice of convective vortices have formed in the forced cells.

Figure 3: Bifurcation curves of
${\langle}{|}{\omega}{|}{\rangle}(\varepsilon)$ near onset. The circles
represent the data of the reference cell. The squares and triangles represent
respectively data for the square-patterned cell with $\lambda{=}14.14$ mm, and
the hexagon-patterned cell with $\lambda{=}17.32$ mm. Error bars denote the
standard deviation. The dotted line shows the fitted square-root law for the
reference cell, and the solid lines are the predicted imperfect bifurcation
curves for the forced cells. Inset panels: time-averaged vorticity fields for
$\varepsilon{\approx}{-}0.1$ of the three cells. Data for
$\Omega{=}1.12{\times}10^{4}$.

In an effort to understand the $\varepsilon$-dependence of
${\langle}{|}{\omega}{|}{\rangle}$ near onset, we propose a phenomenological
Ginzburg-Landau-like model for the convection amplitude $A_{j}$ of rotating
RBC Scheel (2007); Scheel et al. (2010) in the presence of external periodic
forcing Kelly and Pal (1978); Coullet (1986); McCoy (2007); Seiden et al.
(2008)

${\partial}_{t}A_{j}{=}{\varepsilon}A_{j}{+}{\xi}_{0}^{2}{\nabla}^{2}A_{j}{-}\sum_{i{=}1}^{n}g_{0}^{ij}|A_{i}|^{2}A_{j}{+}g_{2}^{j}{\delta}_{j}A_{j}^{{\ast}m{-}1}.$
(1)

Here $g_{0}^{ij}$ is the nonlinear coupling coefficient between the Fourier
modes $i$ and $j$ of the flow pattern and $n$ is the number of dominant modes.
$g_{2}^{j}$ is an imperfection coefficient. $\delta_{j}$ represents the
strength of the external forcing. The integer $m$ denotes the degree of
resonance and we consider here resonant forcing
($k_{\mathrm{f}}{\approx}k_{0}$ and $m{=}1$). $\xi_{0}$ is a spatial gradient
coefficient. In view of the symmetry of the flow patterns shown in Figs. 1b
and 1c, we set $g_{0}^{ij}{=}g_{0}$ for any two coupling modes as constants
and $g_{2}^{j}{=}g_{2}$. We consider a stationary solution for the near-onset
flow regime. Through summation of all modes Eq. (1) is reduced to an amplitude
equation:
${(\varepsilon{+}\varepsilon_{0})}A{-}g_{0}{|}A{|}^{2}A/n{+}g_{2}{\delta}{=}0$,
with $\delta{=}\sum_{j{=}1}^{n}{\delta}_{j}$. The offset $\varepsilon_{0}$
represents a shift of $\mathrm{Ra}_{\mathrm{c}}$ since strictly
$\vec{k}_{\mathrm{f}}{\neq}\vec{k}_{0}$ Cross (1980), and since the local
temperature gradient increases slightly over the bottom texture McCoy (2007);
Seiden et al. (2008). The amplitude $A{=}\sum_{j{=}1}^{n}A_{j}$, is related to
the observed mean vorticity modulus ${\langle}{|}{\omega}{|}{\rangle}$ through
a scale factor $S$, $A{=}S{\langle}{|}{\omega}{|}{\rangle}$ SM . We show in
Fig. 3 the theoretical predictions
${\langle}{|}{\omega}{|}{\rangle}{=}\sqrt{\varepsilon/g_{0}}S^{{-}1}$ for the
reference cell ($\delta{=}0$), and
${\langle}{|}{\omega}{|}{\rangle}(\varepsilon)$ for the two forced cells that
fulfills a cubic equation. Both the experimental data and the theoretical
curve show clearly the signature of a forward bifurcation near
$\varepsilon{=}0$ when the external forcing is absent. Meanwhile the
pronounced rounding of the transition in the two forced cells indicates an
imperfect bifurcation. The parameter values obtained from fitting the
bifurcation data can be interpreted as well by the model SM . The agreement
between the experimental and theoretical results suggest that the physics of
external modulation near onset of rotating convection can be well described by
the present Ginzburg-Landau-like model.

Figure 4: (a-d) Trajectories of cyclones (red) and anticyclones (blue)
overlying a square array of cylinders (gray circles). The dark (light) circle
on each trajectory indicate where the vortex appears (terminates). Results for
$\Omega{=}1.12{\times}10^{4}$, $\lambda{=}14.14$ mm and $\varepsilon{=}0.49$
for (a), 1.74 (b), 4.18 (c) and 9.31 (d). (e, f) Mean-square-displacements
${\delta}\vec{r}^{2}(t)$ of cyclones (e) and anticyclones (f) in the square-
patterned cell. Open circles show the data of the reference cell. Insets in
(e) and (f) show the diffusion coefficient $D$ of vortex motion in the
reference cell (open circles) and the forced cell (solid squares).

The convection pattern becomes time-dependent and the vortices are meandering
horizontally when $\varepsilon$ is far above the onset. To explore the impact
of the bottom texture on the flow field far from onset, we examine here the
motions of the vortices with varying buoyancy strength. Figures 4a-4d show the
trajectories of the vortices, overlying a gray texture that represents a
square array of cylinders on the bottom surface. Near onset
($\varepsilon{=}0.49$), the motion of cyclones is severely restricted within
the fluid domain over the raised cylinders. Meanwhile anticyclones appear at
their stable positions, i.e., the middle points of every four adjacent
cylinders, so as to maintain the largest distance from its nearest cyclones
(Fig. 4a). With increased buoyancy forcing ($\varepsilon{=}1.74$), while the
cyclones are still confined above the raised cylinders, the anticyclones may
move in between the stable positions, with their chaotic trajectories forming
a web-shape domain of anticyclonic motion (Fig. 4b). A further increase in
$\varepsilon$ ($\varepsilon{=}4.18$) releases the constraint of vortex motion
as shown in Fig. 4c. When a sufficiently large buoyancy forcing
($\varepsilon{=}9.31$) is applied, the trajectories of both cyclones and
anticyclones cover nearly the full fluid domain as the effect of external
forcing diminishes (Fig. 4d), and we find that the stochastic vortex motion
Chong et al. (2020); Ding et al. (2021) recovers.

To characterize the statistical behavior of the vortices, we compute their
mean-square displacements,
${\delta}\vec{r}^{2}(t){=}{\langle}(\vec{r}(\tau{+}t){-}\vec{r}(\tau))^{2}{\rangle}_{\tau}$,
using the trajectories $\vec{r}(t)$ obtained above. Figures 4e and 4f show the
results for cyclones and anticyclones, respectively. With strong buoyancy
forcing the data curves of ${\delta}\vec{r}^{2}(t)$ for cyclones in the forced
cell resemble that of the reference cell (Fig. 4e), which reveal that the
vortex motion undergoes a transition from ballistic to diffusive motion as
time increases Chong et al. (2020). When $\varepsilon$ decreases
${\delta}\vec{r}^{2}(t)$ of the forced cell decreases progressively,
signifying that the restriction effect of the cyclonic motion due to the
external forcing becomes significant. A similar $\varepsilon$-dependence of
${\delta}\vec{r}^{2}(t)$ is found in the anticyclonic data (Fig. 4f), although
at low $\varepsilon$ a less reduction of ${\delta}\vec{r}^{2}(t)$ occurs, and
one observes a relatively larger domain of anticyclonic motion (Fig. 4a-4c).
We determine the diffusion coefficient $D$ of vortex motion, using the Green-
Kubo formula Green (1954); Kubo (1957); Morriss and Evans (2007); Hansen and
McDonald (2013):
$D{=}\int_{0}^{\infty}{\langle}\vec{u}(0){\cdot}\vec{u}(t){\rangle}dt$, with
$\vec{u}(t)$ being the velocity of vortex motion. Results for the forced cell
are compared with the reference cell in the insets. We see that with large
$\varepsilon$ ($\varepsilon{\geq}5.0$) data for the two cells overlap. With
decreasing $\varepsilon$, $D$ decreases more rapidly in the forced cell. When
$\varepsilon{\approx}1.0$, $D$ of anticyclonic (cyclonic) motion in the forced
cell becomes one (two) order in magnitudes less than that in the reference
cell. Further decrease in $\varepsilon$ leads to a stationary vortex
distribution in the forced cell as shown in Fig. 1c.

The flow pattern in rapidly rotating convection is characterized by columnar
vortices that undergo stochastic horizontal motion. We have shown that when a
periodically topographic structure is introduced on the heated surface, the
vortex motion can be strictly controlled to form stationary convection
patterns with prescribed symmetries. We demonstrate that the new patterns
arise through a dynamical process of imperfect bifurcation with the effects of
external forcing well described by a Ginzburg-Landau-like model. In rapidly
rotating convection it is predicted that heat and mass are transported mainly
by the columnar vortices Veronis (1959); Julien et al. (1999); King and Aurnou
(2012). Our experimental findings of a wide parameter regime to manipulate
these vortices through topological forcing may thus enable new experimental
approaches to examine and exploit local heat and fluid transports in rotating
flows. The richness of patterns and vortex dynamics in modulated rotating
convection observed in this study may stimulate further theoretical and
numerical investigations, and will contribute to understanding of pattern
formation in non-equilibrium systems constrained by nonuniform boundary
conditions.

This work is supported by the Fundamental Research Funds for the Central
Universities, and the National Science Foundation of China under Grant Nos.
92152105 and 11772235.

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