text: Achievable Rate Regions for Cooperative Relay Broadcast Channels with Feedback Youlong Wu Lehrstuhl fu¨r Nachrichtentechnik Technische Universita¨t Mu¨nchen, Germany youlong.wu@tum.de Abstract— Achievable rate regions for cooperative relay broad- rate-limited feedback link, and meanwhile, acts as a relay to cast channels with rate-limited feedback are proposed. Speciﬁ- transmit cooperative information to the other receiver. cally, we consider two-receiver memoryless broadcast channels In the ﬁrst work, we ﬁrst study the partially coopera- where each receiver sends feedback signals to the transmitter tive RBC with one-sided feedback (only one receiver sends through a noiseless and rate-limited feedback link, meanwhile, acts as a relay to transmit cooperative information to the other feedback signals and relays cooperative information to the receiver. It’s shown that the proposed rate regions improve on other receiver). We proposed a new coding scheme (Scheme the known regions that consider either relaying cooperation or 1) based on block-Markov coding, Marton’s coding [12], feedback communication, but not both. partial decode-forward [13] and compress-forward strategies [13]. Speciﬁcally, in each block, the transmitter uses Mar- I. INTRODUCTION ton’s coding to send the source messages and forward the feedback message. The receiver who acts as relay performs Relay broadcast channels (RBCs) describe communication combined partial decode-forward and compress-forward, and networks where the transmitter sends information to a set of sends the compression message as feedback information. The receivers with the help of relaying communication. In [1], other Receiver uses backward decoding to jointly decode its [2], the dedicated-relay broadcast channel (BC) model was private message and the compression message. It is shown that studied, where a relay node was introduced to the original when feedback rate is sufﬁciently large, our coding scheme BC to assist the cooperation between two receivers. Another strictly improves on Liang and Kramer’s region [4], which is RBC model, called cooperative RBC model was studied in [3], tight for the semideterministic partially cooperative RBCs and [4], where each receiver acts as a relay and sends cooperative orthogonal partially cooperative RBCs. information to the other receiver. It was shown that even In the second work, we study the fully cooperative RBCs partially cooperation (only one receiver relays cooperative with two-sided feedback (both receiver send feedback signals information) still improves on the capacity region of original and relay cooperative information). Two block-Markov coding BC. schemes (Scheme 2A and 2B) are proposed based on Scheme In a different line of work, many studies have been done on 1. Speciﬁcally, in each block, the transmitter uses Marton’s memoryless BCs with feedback, where the receivers send feed- coding to send the source messages and forward the feedback back signals to the transmitter through feedback links. In [5], it messages sent by both receivers. In Scheme 2A, both receivers shows that feedback cannot increase the capacity region for all apply compress-forward and backward decoding. Scheme 2B physically degraded BCs. The ﬁrst simple example BC where is similar to Scheme 1A except that one of the two receiver feedback increases capacity was presented by Dueck [6]. uses hybrid relaying strategy and sliding-window decoding. Based on Dueck’s idea, Shayevitz and Wigger [7] proposed an The resulting rate regions strictly improve on Wu and Wigger’s achievable region for BCs with generalized feedback. Other region [10, Theorem 1], which shows that feedback strictly achievable regions for BCs with perfect or noisy feedback, increases capacity region for a large class of BCs. have been proposed by Kramer [8] and Venkataramanan and Note that in our coding schemes the transmitter can recon- Pradhan [9]. Most recently, Wu and Wigger [10], [11] showed struct the receivers’ inputs due to a delicate design, which that any positive feedback rate can increase the capacity region allows to superimpose the Marton’s codes on the receivers for a large class of BCs, called strictly essentially less noisy inputs, and thus attains cooperation between the transmitter BCs, unless it is physically degraded. and the receivers. Cooperative RBCs with prefect feedback was investigated This paper is organized as follows. Section describes in [3], where the capacity region was established for the case cooperative RBC with feedback and our main results are of perfect feedback from the receiver to the relay. In this paper, presented in Section . Section 7 compares various achievable we consider the cooperative RBCs with rate-limited feedback rate regions and shows that our regions strictly improve the from the receivers to the transmitter, i.e., each receiver sends known rate regions that consider either relay cooperation or feedback signals to the transmitter through a noiseless and feedback communication, but not both. Sections 16 and 12

text: Achievable

text: Rate

text: Regions

text: for

text: Cooperative

text: Relay

text: Broadcast

text: Channels

text: with

text: Feedback

text: Youlong

text: Wu

text: Lehrstuhl

text: fu¨r

text: Nachrichtentechnik

text: Technische

text: Universita¨t

text: Mu¨nchen,

text: Germany

text: youlong.wu@tum.de

text: Abstract—

text: Achievable

text: rate

text: regions

text: for

text: cooperative

text: relay

text: broad-

text: rate-limited

text: feedback

text: link,

text: and

text: meanwhile,

text: acts

text: as

text: a

text: relay

text: to

text: cast

text: channels

text: with

text: rate-limited

text: feedback

text: are

text: proposed.

text: Speciﬁ-

text: transmit

text: cooperative

text: information

text: to

text: the

text: other

text: receiver.

text: cally,

text: we

text: consider

text: two-receiver

text: memoryless

text: broadcast

text: channels

text: In

text: the

text: ﬁrst

text: work,

text: we

text: ﬁrst

text: study

text: the

text: partially

text: coopera-

text: where

text: each

text: receiver

text: sends

text: feedback

text: signals

text: to

text: the

text: transmitter

text: tive

text: RBC

text: with

text: one-sided

text: feedback

text: (only

text: one

text: receiver

text: sends

text: through

text: a

text: noiseless

text: and

text: rate-limited

text: feedback

text: link,

text: meanwhile,

text: acts

text: as

text: a

text: relay

text: to

text: transmit

text: cooperative

text: information

text: to

text: the

text: other

text: feedback

text: signals

text: and

text: relays

text: cooperative

text: information

text: to

text: the

text: receiver.

text: It’s

text: shown

text: that

text: the

text: proposed

text: rate

text: regions

text: improve

text: on

text: other

text: receiver).

text: We

text: proposed

text: a

text: new

text: coding

text: scheme

text: (Scheme

text: the

text: known

text: regions

text: that

text: consider

text: either

text: relaying

text: cooperation

text: or

text: based

text: on

text: block-Markov

text: coding,

text: Marton’s

text: coding

text: feedback

text: communication,

text: but

text: not

text: both.

text: partial

text: decode-forward

text: and

text: compress-forward

text: strategies

text: Speciﬁcally,

text: in

text: each

text: block,

text: the

text: transmitter

text: uses

text: Mar-

text: I.

text: INTRODUCTION

text: ton’s

text: coding

text: to

text: send

text: the

text: source

text: messages

text: and

text: forward

text: the

text: feedback

text: message.

text: The

text: receiver

text: who

text: acts

text: as

text: relay

text: performs

text: Relay

text: broadcast

text: channels

text: (RBCs)

text: describe

text: communication

text: combined

text: partial

text: decode-forward

text: and

text: compress-forward,

text: and

text: networks

text: where

text: the

text: transmitter

text: sends

text: information

text: to

text: a

text: set

text: of

text: sends

text: the

text: compression

text: message

text: as

text: feedback

text: information.

text: The

text: receivers

text: with

text: the

text: help

text: of

text: relaying

text: communication.

text: In

text: other

text: Receiver

text: uses

text: backward

text: decoding

text: to

text: jointly

text: decode

text: its

text: the

text: dedicated-relay

text: broadcast

text: channel

text: (BC)

text: model

text: was

text: private

text: message

text: and

text: the

text: compression

text: message.

text: It

text: is

text: shown

text: that

text: studied,

text: where

text: a

text: relay

text: node

text: was

text: introduced

text: to

text: the

text: original

text: when

text: feedback

text: rate

text: is

text: sufﬁciently

text: large,

text: our

text: coding

text: scheme

text: BC

text: to

text: assist

text: the

text: cooperation

text: between

text: two

text: receivers.

text: Another

text: strictly

text: improves

text: on

text: Liang

text: and

text: Kramer’s

text: region

text: which

text: is

text: RBC

text: model,

text: called

text: cooperative

text: RBC

text: model

text: was

text: studied

text: in

text: tight

text: for

text: the

text: semideterministic

text: partially

text: cooperative

text: RBCs

text: and

text: where

text: each

text: receiver

text: acts

text: as

text: a

text: relay

text: and

text: sends

text: cooperative

text: orthogonal

text: partially

text: cooperative

text: RBCs.

text: information

text: to

text: the

text: other

text: receiver.

text: It

text: was

text: shown

text: that

text: even

text: In

text: the

text: second

text: work,

text: we

text: study

text: the

text: fully

text: cooperative

text: RBCs

text: partially

text: cooperation

text: (only

text: one

text: receiver

text: relays

text: cooperative

text: with

text: two-sided

text: feedback

text: (both

text: receiver

text: send

text: feedback

text: signals

text: information)

text: still

text: improves

text: on

text: the

text: capacity

text: region

text: of

text: original

text: and

text: relay

text: cooperative

text: information).

text: Two

text: block-Markov

text: coding

text: BC.

text: schemes

text: (Scheme

text: and

text: are

text: proposed

text: based

text: on

text: Scheme

text: In

text: a

text: different

text: line

text: of

text: work,

text: many

text: studies

text: have

text: been

text: done

text: on

text: Speciﬁcally,

text: in

text: each

text: block,

text: the

text: transmitter

text: uses

text: Marton’s

text: memoryless

text: BCs

text: with

text: feedback,

text: where

text: the

text: receivers

text: send

text: feed-

text: coding

text: to

text: send

text: the

text: source

text: messages

text: and

text: forward

text: the

text: feedback

text: back

text: signals

text: to

text: the

text: transmitter

text: through

text: feedback

text: links.

text: In

text: it

text: messages

text: sent

text: by

text: both

text: receivers.

text: In

text: Scheme

text: both

text: receivers

text: shows

text: that

text: feedback

text: cannot

text: increase

text: the

text: capacity

text: region

text: for

text: all

text: apply

text: compress-forward

text: and

text: backward

text: decoding.

text: Scheme

text: physically

text: degraded

text: BCs.

text: The

text: ﬁrst

text: simple

text: example

text: BC

text: where

text: is

text: similar

text: to

text: Scheme

text: except

text: that

text: one

text: of

text: the

text: two

text: receiver

text: feedback

text: increases

text: capacity

text: was

text: presented

text: by

text: Dueck

text: uses

text: hybrid

text: relaying

text: strategy

text: and

text: sliding-window

text: decoding.

text: Based

text: on

text: Dueck’s

text: idea,

text: Shayevitz

text: and

text: Wigger

text: proposed

text: an

text: The

text: resulting

text: rate

text: regions

text: strictly

text: improve

text: on

text: Wu

text: and

text: Wigger’s

text: achievable

text: region

text: for

text: BCs

text: with

text: generalized

text: feedback.

text: Other

text: region

text: Theorem

text: which

text: shows

text: that

text: feedback

text: strictly

text: achievable

text: regions

text: for

text: BCs

text: with

text: perfect

text: or

text: noisy

text: feedback,

text: increases

text: capacity

text: region

text: for

text: a

text: large

text: class

text: of

text: BCs.

text: have

text: been

text: proposed

text: by

text: Kramer

text: and

text: Venkataramanan

text: and

text: Note

text: that

text: in

text: our

text: coding

text: schemes

text: the

text: transmitter

text: can

text: recon-

text: Pradhan

text: Most

text: recently,

text: Wu

text: and

text: Wigger

text: showed

text: struct

text: the

text: receivers’

text: inputs

text: due

text: to

text: a

text: delicate

text: design,

text: which

text: that

text: any

text: positive

text: feedback

text: rate

text: can

text: increase

text: the

text: capacity

text: region

text: allows

text: to

text: superimpose

text: the

text: Marton’s

text: codes

text: on

text: the

text: receivers

text: for

text: a

text: large

text: class

text: of

text: BCs,

text: called

text: strictly

text: essentially

text: less

text: noisy

text: inputs,

text: and

text: thus

text: attains

text: cooperation

text: between

text: the

text: transmitter

text: BCs,

text: unless

text: it

text: is

text: physically

text: degraded.

text: and

text: the

text: receivers.

text: Cooperative

text: RBCs

text: with

text: prefect

text: feedback

text: was

text: investigated

text: This

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: describes

text: in

text: where

text: the

text: capacity

text: region

text: was

text: established

text: for

text: the

text: case

text: cooperative

text: RBC

text: with

text: feedback

text: and

text: our

text: main

text: results

text: are

text: of

text: perfect

text: feedback

text: from

text: the

text: receiver

text: to

text: the

text: relay.

text: In

text: this

text: paper,

text: presented

text: in

text: Section

text: Section

text: compares

text: various

text: achievable

text: we

text: consider

text: the

text: cooperative

text: RBCs

text: with

text: rate-limited

text: feedback

text: rate

text: regions

text: and

text: shows

text: that

text: our

text: regions

text: strictly

text: improve

text: the

text: from

text: the

text: receivers

text: to

text: the

text: transmitter,

text: i.e.,

text: each

text: receiver

text: sends

text: known

text: rate

text: regions

text: that

text: consider

text: either

text: relay

text: cooperation

text: or

text: feedback

text: signals

text: to

text: the

text: transmitter

text: through

text: a

text: noiseless

text: and

text: feedback

text: communication,

text: but

text: not

text: both.

text: Sections

text: and


text: 2 actions contain second order derivatives acting on the scalar ﬁeld. They have very attractive phenomenological and theoretical features. In particular, they admit self-accelerating solutions that can explain the phenomenon of dark energy, and at the same time exhibit screening mechanisms, which ensure consistency with the tight bounds on small scale deviations from GR. Once Galileons are covariantised by naively replacing partial derivatives by covariant derivatives, the theory leads to third order derivatives in the equations of motion. A way to avoid them is to introduce counter terms, in the form of non-minimal couplings to gravity [5]. It was then shown that these theories can be further extended by including free functions of the scalar ﬁeld and its kinetic term [6]. Remarkably, it was found that this theory was already proposed in 1974 by Horndeski [7, 8] and this theory is now called the Horndeski theory [9]. The basic strategy to construct the Horndeski theory is to ensure that the equations of motion are of at most second order in derivatives. If the equations of motion contain higher order time derivatives, this generally leads to a ghost instability due to the Ostrogradsky theorem1. However, it was suggested that – despite the appearance of higher order derivatives – it is still possible to avoid the Ostrogradsky ghost if the theory contains a hidden constraint and the equations of motion can be recast into a second order form, for example by ﬁeld redeﬁnitions. This possibility was studied in detail by Ref. [11–14]. In Ref. [11, 12], the authors constructed theories in the so-called unitary gauge where the scalar ﬁeld depends only on time, thus allowing it to be used as a clock. It was then found that there are two more terms with two arbitrary functions of the scalar ﬁeld and its kinetic term that can be added to the Horndeski theory. Remarkably, these two terms can be obtained by naively covariantising the ﬂat space Galileon in the form written with the anti-symmetric Levi-Civita tensor. Although the equations of motion contain third order time derivatives, it was argued that they do not lead to the Ostrogradsky instability, since constraints remove this additional degree of freedom. These terms are dubbed ‘beyond Horndeski’. There still remains the question of whether beyond Horndeski can be obtained from Horndeski by a suitable transformation of the metric. It is indeed possible to perform generalised conformal and disformal transformations [15], that depend on the scalar ﬁeld and its kinetic term, on Horndeski and obtain a theory which contains higher order derivatives [14, 16]. As long as the transformation is invertible, it does not change the physics. Thus, despite the appearance of higher order derivatives, the resultant theory propagates the same number of degrees of freedom as Horndeski. However, in this case, the theory can be mapped into Horndeski, so unless we introduce a coupling to matter, 1 See Ref. [10] for a review. ##LTLine##

label: paragraph

text: actions

text: contain

text: second

text: order

text: derivatives

text: acting

text: on

text: the

text: scalar

text: ﬁeld.

text: They

text: have

text: very

text: attractive

text: phenomenological

text: and

text: theoretical

text: features.

text: In

text: particular,

text: they

text: admit

text: self-accelerating

text: solutions

text: that

text: can

text: explain

text: the

text: phenomenon

text: of

text: dark

text: energy,

text: and

text: at

text: the

text: same

text: time

text: exhibit

text: screening

text: mechanisms,

text: which

text: ensure

text: consistency

text: with

text: the

text: tight

text: bounds

text: on

text: small

text: scale

text: deviations

text: from

text: GR.

text: Once

text: Galileons

text: are

text: covariantised

text: by

text: naively

text: replacing

text: partial

text: derivatives

text: by

text: covariant

text: derivatives,

text: the

text: theory

text: leads

text: to

text: third

text: order

text: derivatives

text: in

text: the

text: equations

text: of

text: motion.

text: A

text: way

text: to

text: avoid

text: them

text: is

text: to

text: introduce

text: counter

text: terms,

text: in

text: the

text: form

text: of

text: non-minimal

text: couplings

text: to

text: gravity

text: It

text: was

text: then

text: shown

text: that

text: these

text: theories

text: can

text: be

text: further

text: extended

text: by

text: including

text: free

text: functions

text: of

text: the

text: scalar

text: ﬁeld

text: and

text: its

text: kinetic

text: term

text: Remarkably,

text: it

text: was

text: found

text: that

text: this

text: theory

text: was

text: already

text: proposed

text: in

text: by

text: Horndeski

text: and

text: this

text: theory

text: is

text: now

text: called

text: the

text: Horndeski

text: theory

text: The

text: basic

text: strategy

text: to

text: construct

text: the

text: Horndeski

text: theory

text: is

text: to

text: ensure

text: that

text: the

text: equations

text: of

text: motion

text: are

text: of

text: at

text: most

text: second

text: order

text: in

text: derivatives.

text: If

text: the

text: equations

text: of

text: motion

text: contain

text: higher

text: order

text: time

text: derivatives,

text: this

text: generally

text: leads

text: to

text: a

text: ghost

text: instability

text: due

text: to

text: the

text: Ostrogradsky

text: However,

text: it

text: was

text: suggested

text: that

text: despite

text: the

text: appearance

text: of

text: higher

text: order

text: derivatives

text: it

text: is

text: still

text: possible

text: to

text: avoid

text: the

text: Ostrogradsky

text: ghost

text: if

text: the

text: theory

text: contains

text: a

text: hidden

text: constraint

text: and

text: the

text: equations

text: of

text: motion

text: can

text: be

text: recast

text: into

text: a

text: second

text: order

text: form,

text: for

text: example

text: by

text: ﬁeld

text: redeﬁnitions.

text: This

text: possibility

text: was

text: studied

text: in

text: detail

text: by

text: Ref.

text: In

text: Ref.

text: the

text: authors

text: constructed

text: theories

text: in

text: the

text: so-called

text: unitary

text: gauge

text: where

text: the

text: scalar

text: ﬁeld

text: depends

text: only

text: on

text: time,

text: thus

text: allowing

text: it

text: to

text: be

text: used

text: as

text: a

text: clock.

text: It

text: was

text: then

text: found

text: that

text: there

text: are

text: two

text: more

text: terms

text: with

text: two

text: arbitrary

text: functions

text: of

text: the

text: scalar

text: ﬁeld

text: and

text: its

text: kinetic

text: term

text: that

text: can

text: be

text: added

text: to

text: the

text: Horndeski

text: theory.

text: Remarkably,

text: these

text: two

text: terms

text: can

text: be

text: obtained

text: by

text: naively

text: covariantising

text: the

text: ﬂat

text: space

text: Galileon

text: in

text: the

text: form

text: written

text: with

text: the

text: anti-symmetric

text: Levi-Civita

text: tensor.

text: Although

text: the

text: equations

text: of

text: motion

text: contain

text: third

text: order

text: time

text: derivatives,

text: it

text: was

text: argued

text: that

text: they

text: do

text: not

text: lead

text: to

text: the

text: Ostrogradsky

text: instability,

text: since

text: constraints

text: remove

text: this

text: additional

text: degree

text: of

text: freedom.

text: These

text: terms

text: are

text: dubbed

text: ‘beyond

text: Horndeski’.

text: There

text: still

text: remains

text: the

text: question

text: of

text: whether

text: beyond

text: Horndeski

text: can

text: be

text: obtained

text: from

text: Horndeski

text: by

text: a

text: suitable

text: transformation

text: of

text: the

text: metric.

text: It

text: is

text: indeed

text: possible

text: to

text: perform

text: generalised

text: conformal

text: and

text: disformal

text: transformations

text: that

text: depend

text: on

text: the

text: scalar

text: ﬁeld

text: and

text: its

text: kinetic

text: term,

text: on

text: Horndeski

text: and

text: obtain

text: a

text: theory

text: which

text: contains

text: higher

text: order

text: derivatives

text: As

text: long

text: as

text: the

text: transformation

text: is

text: invertible,

text: it

text: does

text: not

text: change

text: the

text: physics.

text: Thus,

text: despite

text: the

text: appearance

text: of

text: higher

text: order

text: derivatives,

text: the

text: resultant

text: theory

text: propagates

text: the

text: same

text: number

text: of

text: degrees

text: of

text: freedom

text: as

text: Horndeski.

text: However,

text: in

text: this

text: case,

text: the

text: theory

text: can

text: be

text: mapped

text: into

text: Horndeski,

text: so

text: unless

text: we

text: introduce

text: a

text: coupling

text: to

text: matter,

text: See

text: Ref.

text: for

text: a

text: review.

text: ##LTLine##


text: 16 L L L H ¯ [G¯ ] = H [G ] + BH [D ] , (61) 4 4 4 4 4 4 L L L H ¯ [G¯ ] = H [G ] + BH [D ] ; (62) 5 5 5 5 5 5 where (cid:18) (cid:19) Γ − 2XG¯ G = G¯ (1 + XΓ)1/2 ,(cid:90) D = X G¯ 4X¯ , (63) 4 4 4 4 (1 + XΓ)1/2 1 + XΓ G¯ G¯ (Γ + XΓ ) XG¯ Γ G = 5 + 5 X dX , D = 5X¯ X . (64) 5 5 (1 + XΓ)1/2 2(1 + XΓ)3/2 3(1 + XΓ)5/2 On the other hand, beyond Horndeski is mapped into itself only, with a diﬀerent overall function L L BH ¯ [F¯ ] = BH [F ] , (65) 4 4 4 4 L L BH ¯ [F¯ ] = BH [F ] ; (66) 5 5 5 5 where F¯ F¯ 4 5 F = , F = . (67) 4 5 (1 + XΓ)5/2 (1 + XΓ)7/2 From these results, we can easily see that the combination of Horndeski and beyond Horndeski of the same order can be mapped to Horndeski. Indeed L L L H ¯ [G¯ ] + BH ¯ [F¯ ] = H [G ] , (68) 4 4 4 4 4 4 L L L H ¯ [G¯ ] + BH ¯ [F¯ ] = H [G ] , (69) 5 5 5 5 5 5 for a disformal transformation Γ(X) which satisﬁes [12] ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: L

label: equation

text: L

text: L

text: H

text: [G¯

text: H

text: [G

text: BH

text: [D

text: L

text: L

text: L

text: H

text: [G¯

text: H

text: [G

text: BH

text: [D

text: where

text: Γ

text: G

text: G¯

text: D

text: X

text: G¯

text: XΓ

text: G¯

text: G¯

text: (Γ

text: XΓ

text: XG¯

text: Γ

text: G

text: X

text: dX

text: D

text: X

text: On

text: the

text: other

text: hand,

text: beyond

text: Horndeski

text: is

text: mapped

text: into

text: itself

text: only,

text: with

text: a

text: diﬀerent

text: overall

text: function

text: L

text: L

text: BH

text: [F¯

text: BH

text: [F

text: L

text: L

text: BH

text: [F¯

text: BH

text: [F

text: where

text: F¯

text: F¯

text: F

text: F

text: From

text: these

text: results,

text: we

text: can

text: easily

text: see

text: that

text: the

text: combination

text: of

text: Horndeski

text: and

text: beyond

text: Horndeski

text: of

text: the

text: same

text: order

text: can

text: be

text: mapped

text: to

text: Horndeski.

text: Indeed

text: L

text: L

text: L

text: H

text: [G¯

text: BH

text: [F¯

text: H

text: [G

text: L

text: L

text: L

text: H

text: [G¯

text: BH

text: [F¯

text: H

text: [G

text: for

text: a

text: disformal

text: transformation

text: Γ(X)

text: which

text: satisﬁes

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 19 FIG. 1: Summary of the results on the combination of Horndeski and beyond Horndeski terms. H indicates that the theory can be mapped to Horndeski by a disformal tranformation while BH indicates that the disformal transformation maps it to beyond Horndeski. Crosses indicate theories that cannot be joined together without loosing the linear primary constraint. ﬁeld kinetic term, without ﬁxing any gauge. The results of this analysis are summarised in ﬁgure 1. We conﬁrmed that Horndeski is disformally mapped to Horndeski plus beyond Horndeski of the same order [12]. On the other hand, beyond Horndeski is mapped to itself by the generalised disformal transformation. Thus, in the absence of coupling with matter, the mixture of Horndeski and beyond Horndeski of the same order is nothing but Horndeski itself. We checked that the redeﬁnition of the extrinsic curvature that removes the mixing term can be derived from the disformal transformation only when there are both Horndeski and beyond Horndeski together. On the other hand, the disformal transformation that maps Horndeski plus beyond Horndeski to Horndeski is singular precisely in the limit that would leave the theory of beyond Horndeski alone. This is consistent with the fact that beyond Horndeski is mapped to beyond Horndeski itself. In this case the transformation to remove the mixing term cannot be derived from the disformal transformation. This indicates that beyond Horndeski is disformally disconnected from Horndenski. The fact that beyond Horndeski cannot be mapped into Horndeski, makes this theory a covariant scalar-tensor theory that can be recast into a second order system which is not Horndeski, therefore it should be Lorentz breaking. It is also interesting that the primary constraint is preserved only when we include a Horndeski term of the same order, but it is spoiled when a diﬀerent order Horndeski term is added. In the ﬁrst case however the theory is just Horndeski up to a generalised disformal transformation. ##LTFigure## ##LTFigure##

text: FIG.

text: Summary

label: caption

text: of

text: the

text: results

text: on

text: the

text: combination

text: of

text: Horndeski

text: and

text: beyond

text: Horndeski

text: terms.

text: H

text: indicates

text: that

text: the

text: theory

text: can

text: be

text: mapped

text: to

text: Horndeski

text: by

text: a

text: disformal

text: tranformation

text: while

text: BH

text: indicates

text: that

text: the

text: disformal

text: transformation

text: maps

text: it

text: to

text: beyond

text: Horndeski.

text: Crosses

text: indicate

text: theories

text: that

text: cannot

text: be

text: joined

text: together

text: without

text: loosing

text: the

text: linear

text: primary

text: constraint.

text: ﬁeld

text: kinetic

text: term,

text: without

text: ﬁxing

text: any

text: gauge.

text: The

text: results

text: of

text: this

text: analysis

text: are

text: summarised

text: in

text: ﬁgure

text: We

text: conﬁrmed

text: that

text: Horndeski

text: is

text: disformally

text: mapped

text: to

text: Horndeski

text: plus

text: beyond

text: Horndeski

text: of

text: the

text: same

text: order

text: On

text: the

text: other

text: hand,

text: beyond

text: Horndeski

text: is

text: mapped

text: to

text: itself

text: by

text: the

text: generalised

text: disformal

text: transformation.

text: Thus,

text: in

text: the

text: absence

text: of

text: coupling

text: with

text: matter,

text: the

text: mixture

text: of

text: Horndeski

text: and

text: beyond

text: Horndeski

text: of

text: the

text: same

text: order

text: is

text: nothing

text: but

text: Horndeski

text: itself.

text: We

text: checked

text: that

text: the

text: redeﬁnition

text: of

text: the

text: extrinsic

text: curvature

text: that

text: removes

text: the

text: mixing

text: term

text: can

text: be

text: derived

text: from

text: the

text: disformal

text: transformation

text: only

text: when

text: there

text: are

text: both

text: Horndeski

text: and

text: beyond

text: Horndeski

text: together.

text: On

text: the

text: other

text: hand,

text: the

text: disformal

text: transformation

text: that

text: maps

text: Horndeski

text: plus

text: beyond

text: Horndeski

text: to

text: Horndeski

text: is

text: singular

text: precisely

text: in

text: the

text: limit

text: that

text: would

text: leave

text: the

text: theory

text: of

text: beyond

text: Horndeski

text: alone.

text: This

text: is

text: consistent

text: with

text: the

text: fact

text: that

text: beyond

text: Horndeski

text: is

text: mapped

text: to

text: beyond

text: Horndeski

text: itself.

text: In

text: this

text: case

text: the

text: transformation

text: to

text: remove

text: the

text: mixing

text: term

text: cannot

text: be

text: derived

text: from

text: the

text: disformal

text: transformation.

text: This

text: indicates

text: that

text: beyond

text: Horndeski

text: is

text: disformally

text: disconnected

text: from

text: Horndenski.

text: The

text: fact

text: that

text: beyond

text: Horndeski

text: cannot

text: be

text: mapped

text: into

text: Horndeski,

text: makes

text: this

text: theory

text: a

text: covariant

text: scalar-tensor

text: theory

text: that

text: can

text: be

text: recast

text: into

text: a

text: second

text: order

text: system

text: which

text: is

text: not

text: Horndeski,

text: therefore

text: it

text: should

text: be

text: Lorentz

text: breaking.

text: It

text: is

text: also

text: interesting

text: that

text: the

text: primary

text: constraint

text: is

text: preserved

text: only

text: when

text: we

text: include

text: a

text: Horndeski

text: term

text: of

text: the

text: same

text: order,

text: but

text: it

text: is

text: spoiled

text: when

text: a

text: diﬀerent

text: order

text: Horndeski

text: term

text: is

text: added.

text: In

text: the

text: ﬁrst

text: case

text: however

text: the

text: theory

text: is

text: just

text: Horndeski

text: up

text: to

text: a

text: generalised

text: disformal

text: transformation.

text: ##LTFigure##

text: ##LTFigure##


text: 21 Clearly in (A3) there is an additional piece proportional to V∗ that it is not present for the disformal transformation. This is the key term. When we add together beyond Horndeski + Horndeski of the same order, using (70)–(71) in the analogous equation of (A3) coming from (58)–(59), this extra piece vanishes and the transformation reduces to (A2). On the other hand with beyond Horndeski alone this extra term survives making the two transformations diﬀerent. We can therefore conclude that, since the transformation that is able to recast beyond Horndeski into a second order theory is not a disformal transformation, beyond Horndeski is equivalent to a Lorentz breaking second order theory. Notice that the singular limit encountered in section , i.e. − Γ = 1/X, manifests also here: indeed for this value the extra piece does not cancel out as one 4,5 would expect. [1] K. Koyama, arXiv:1504.04623 [astro-ph.CO]. [2] C. Brans and R. H. Dicke, Phys. Rev. 124 (1961) 925. [3] Y. Fujii and K. Maeda, “The scalar-tensor theory of gravitation,” Cambridge University Press, 2003. [4] A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197 [hep-th]]. [5] C. Deﬀayet, G. Esposito-Farese and A. Vikman, Phys. Rev. D 79 (2009) 084003 [arXiv:0901.1314 [hep-th]]. [6] C. Deﬀayet, S. Deser and G. Esposito-Farese, Phys. Rev. D 80 (2009) 064015 [arXiv:0906.1967 [gr-qc]]. [7] C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saﬃn, Phys. Rev. Lett. 108 (2012) 051101 [arXiv:1106.2000 [hep-th]]. [8] T. Kobayashi, M. Yamaguchi and J. Yokoyama, Prog. Theor. Phys. 126 (2011) 511 [arXiv:1105.5723 [hep-th]]. [9] G. W. Horndeski, Int. J. Theor. Phys. 10 (1974) 363. [10] R. P. Woodard, Lect. Notes Phys. 720 (2007) 403, [astro-ph/0601672]. [11] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Phys. Rev. Lett. 114 (2015) 21, 211101 [arXiv:1404.6495 [hep-th]]. [12] J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, JCAP 1502 (2015) 018 [arXiv:1408.1952 [astro- ph.CO]]. [13] X. Gao, Phys. Rev. D 90 (2014) 081501 [arXiv:1406.0822 [gr-qc]]. · [14] M. Zumalac rregui and J. Garc`Ia-Bellido, Phys. Rev. D 89 (2014) 064046 [arXiv:1308.4685 [gr-qc]]. [15] J. D. Bekenstein, Phys. Rev. D 48 (1993) 3641 [gr-qc/9211017]. [16] G. DomE¨nech, S. Mukohyama, R. Namba, A. Naruko, R. Saitou and Y. Watanabe, Phys. Rev. D 92 (2015) 8, 084027 [arXiv:1507.05390 [hep-th]]. [17] D. Langlois and K. Noui, arXiv:1510.06930 [gr-qc]. ##LTLine## ##LTLine## ##LTLine##

text: Clearly

text: in

text: there

text: is

text: an

text: additional

text: piece

text: proportional

text: to

text: V∗

text: that

text: it

text: is

text: not

text: present

text: for

text: the

text: disformal

text: transformation.

text: This

text: is

text: the

text: key

text: term.

text: When

text: we

text: add

text: together

text: beyond

text: Horndeski

text: Horndeski

text: of

text: the

text: same

text: order,

text: using

text: in

text: the

text: analogous

text: equation

text: of

text: coming

text: from

text: this

text: extra

text: piece

text: vanishes

text: and

text: the

text: transformation

text: reduces

text: to

text: On

text: the

text: other

text: hand

text: with

text: beyond

text: Horndeski

text: alone

text: this

text: extra

text: term

text: survives

text: making

text: the

text: two

text: transformations

text: diﬀerent.

text: We

text: can

text: therefore

text: conclude

text: that,

text: since

text: the

text: transformation

text: that

text: is

text: able

text: to

text: recast

text: beyond

text: Horndeski

text: into

text: a

text: second

text: order

text: theory

text: is

text: not

text: a

text: disformal

text: transformation,

text: beyond

text: Horndeski

text: is

text: equivalent

text: to

text: a

text: Lorentz

text: breaking

text: second

text: order

text: theory.

text: Notice

text: that

text: the

text: singular

text: limit

text: encountered

text: in

text: section

text: i.e.

text: Γ

text: manifests

text: also

text: here:

text: indeed

text: for

text: this

text: value

text: the

text: extra

text: piece

text: does

text: not

text: cancel

text: out

text: as

text: one

text: would

text: expect.

text: K.

text: Koyama,

text: [astro-ph.CO].

text: C.

text: Brans

text: and

text: R.

text: H.

text: Dicke,

text: Phys.

text: Rev.

text: Y.

text: Fujii

text: and

text: K.

text: Maeda,

text: “The

text: scalar-tensor

text: theory

text: of

text: gravitation,”

text: Cambridge

text: University

text: Press,

text: A.

text: Nicolis,

text: R.

text: Rattazzi

text: and

text: E.

text: Trincherini,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: C.

text: Deﬀayet,

text: G.

text: Esposito-Farese

text: and

text: A.

text: Vikman,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: C.

text: Deﬀayet,

text: S.

text: Deser

text: and

text: G.

text: Esposito-Farese,

text: Phys.

text: Rev.

text: D

text: [gr-qc]].

text: C.

text: Charmousis,

text: E.

text: J.

text: Copeland,

text: A.

text: Padilla

text: and

text: P.

text: M.

text: Saﬃn,

text: Phys.

text: Rev.

text: Lett.

text: [hep-th]].

text: T.

text: Kobayashi,

text: M.

text: Yamaguchi

text: and

text: J.

text: Yokoyama,

text: Prog.

text: Theor.

text: Phys.

text: [hep-th]].

text: G.

text: W.

text: Horndeski,

text: Int.

text: J.

text: Theor.

text: Phys.

text: R.

text: P.

text: Woodard,

text: Lect.

text: Notes

text: Phys.

text: J.

text: Gleyzes,

text: D.

text: Langlois,

text: F.

text: Piazza

text: and

text: F.

text: Vernizzi,

text: Phys.

text: Rev.

text: Lett.

text: [hep-th]].

text: J.

text: Gleyzes,

text: D.

text: Langlois,

text: F.

text: Piazza

text: and

text: F.

text: Vernizzi,

text: JCAP

text: [astro-

text: ph.CO]].

text: X.

text: Gao,

text: Phys.

text: Rev.

text: D

text: [gr-qc]].

text: M.

text: Zumalac

text: rregui

text: and

text: J.

text: Garc`Ia-Bellido,

text: Phys.

text: Rev.

text: D

text: [gr-qc]].

text: J.

text: D.

text: Bekenstein,

text: Phys.

text: Rev.

text: D

text: G.

text: DomE¨nech,

text: S.

text: Mukohyama,

text: R.

text: Namba,

text: A.

text: Naruko,

text: R.

text: Saitou

text: and

text: Y.

text: Watanabe,

text: Phys.

text: Rev.

text: D

text: [hep-th]].

text: D.

text: Langlois

text: and

text: K.

text: Noui,

text: [gr-qc].

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 Using the expression for the extrinsic curvature (16), this can be written as L M ∇ ∇ BH = XF (φ, X) αβ Aµ Aν, (19) 4 4 µν α β L M ∇ ∇ ∇ BH = XF (φ, X) αβγ Aµ Aν Aρ, (20) 5 5 µνρ α β γ where M M αβ = P αP β , αβγ = P αP βP γ . (21) µν [µ ν] µνρ [µ ν ρ] M The matrix has a crucial property for the arguments that we are going to develop: M M α..γ A Aµ = α..γ A Aρ = 0 . (22) µ...ρ α µ...ρ γ M This property stems from the fact that is built in terms of the projection tensor (15). As we will show in section , this leads to the existence of a primary constraint necessary to avoid the propagation of an extra mode in beyond Horndeski, despite the fact that the equations of motion have higher derivatives. We stress that the unitary gauge used in the literature [12, 22] is a very special choice of gauge where the constant time hypersurface and the constant scalar ﬁeld hypersurface coincide. The analysis in this special gauge could lead to misleading results, as was explicitly pointed out ﬁrst in [18] and then in [17, 19]. For this reason, we will refrain from making this choice in what comes next. III. KINETIC TERMS In order to identify the kinetic terms and carry out the analysis of constraints we need to separate space and time, performing a 3+1 decomposition. We introduce the time vector ﬂow tµ = ∂/∂t decomposed as tµ = N nµ + Nµ , (23) where nµ is the unit normal vector to the t = const. hypersurface, N the lapse function and Nµ the shift vector orthogonal to the normal vector. The constant time hypersurface is then characterised by the following three quantities:

text: Using

text: the

text: expression

text: for

text: the

text: extrinsic

text: curvature

text: this

text: can

text: be

text: written

text: as

text: L

text: M

text: BH

text: XF

text: (φ,

text: X)

text: αβ

text: Aµ

text: Aν,

text: µν

text: α

text: β

text: L

text: M

text: BH

text: XF

text: (φ,

text: X)

text: αβγ

text: Aµ

text: Aν

text: Aρ,

text: µνρ

text: α

text: β

text: γ

text: where

text: M

text: M

text: αβ

text: P

text: αP

text: β

text: αβγ

text: P

text: αP

text: βP

text: γ

text: µν

text: [µ

text: ν]

text: µνρ

text: [µ

text: ν

text: ρ]

text: M

text: The

text: matrix

text: has

text: a

text: crucial

text: property

text: for

text: the

text: arguments

text: that

text: we

text: are

text: going

text: to

text: develop:

text: M

text: M

text: α..γ

text: A

text: Aµ

text: α..γ

text: A

text: Aρ

text: µ...ρ

text: α

text: µ...ρ

text: γ

text: M

text: This

text: property

text: stems

text: from

text: the

text: fact

text: that

text: is

text: built

text: in

text: terms

text: of

text: the

text: projection

text: tensor

text: As

text: we

text: will

text: show

text: in

text: section

text: this

text: leads

text: to

text: the

text: existence

text: of

text: a

text: primary

text: constraint

text: necessary

text: to

text: avoid

text: the

text: propagation

text: of

text: an

text: extra

text: mode

text: in

text: beyond

text: Horndeski,

text: despite

text: the

text: fact

text: that

text: the

text: equations

text: of

text: motion

text: have

text: higher

text: derivatives.

text: We

text: stress

text: that

text: the

text: unitary

text: gauge

text: used

text: in

text: the

text: literature

text: is

text: a

text: very

text: special

text: choice

text: of

text: gauge

text: where

text: the

text: constant

text: time

text: hypersurface

text: and

text: the

text: constant

text: scalar

text: ﬁeld

text: hypersurface

text: coincide.

text: The

text: analysis

text: in

text: this

text: special

text: gauge

text: could

text: lead

text: to

text: misleading

text: results,

text: as

text: was

text: explicitly

text: pointed

text: out

text: ﬁrst

text: in

text: and

text: then

text: in

text: For

text: this

text: reason,

text: we

text: will

text: refrain

text: from

text: making

text: this

text: choice

text: in

text: what

text: comes

text: next.

text: III.

text: KINETIC

text: TERMS

text: In

text: order

text: to

text: identify

text: the

text: kinetic

text: terms

text: and

text: carry

text: out

text: the

text: analysis

text: of

text: constraints

text: we

text: need

text: to

text: separate

text: space

text: and

text: time,

text: performing

text: a

text: decomposition.

text: We

text: introduce

text: the

text: time

text: vector

text: ﬂow

text: tµ

text: ∂/∂t

text: decomposed

text: as

text: tµ

text: N

text: nµ

text: Nµ

text: where

text: nµ

text: is

text: the

text: unit

text: normal

text: vector

text: to

text: the

text: t

text: const.

text: hypersurface,

text: N

text: the

text: lapse

text: function

text: and

text: Nµ

text: the

text: shift

text: vector

text: orthogonal

text: to

text: the

text: normal

text: vector.

text: The

text: constant

text: time

text: hypersurface

text: is

text: then

text: characterised

text: by

text: the

text: following

text: three

text: quantities:


text: A. Paredes, M. Gellert, G. Ru¨diger: Mixing of a passive scalar Mach number becomes unity at Re = 253. In the slow rota- uration for larger Ha is indicated by the results presented in Fig. tion regime, the effective diffusivity hardly changes. The fact 6. It is not yet clear whether for large Ha an upper limit exists ∗ ∗ that even without rotation the ratio D /D is different from zero for Re due to numerical limitations. is because the ﬂow is unstable even for Re = 0. In the fast rota- ∗ tion regime, the ratio D /D increases in a monotonic way until (cid:39) 5. Conclusions it peaks at about Mm 2. Finally, for faster rotation (Mm > 2) the effective diffusivity decays due to the rotational quenching. The inﬂuence of the current-induced instability on the effec- In all cases, however, the effective normalized diffusivity grows tive diffusivity in radial direction of a rotating pinch has been ∗ ∝ with growing Schmidt number so that also here simply D ν studied for different rotation laws. The diffusion equation is without any inﬂuence of the microscopic diffusivity. The miss- numerically solved in a cylindric setup under the inﬂuence of ing factor in this relation is simply given by the curve for Sc = 1 stochastic ﬂuctuations which are due to the magnetic Tayler in- in Fig. 5. stability. The conducting ﬂuid between two rotating cylinders ∗ For all Schmidt numbers, the ratio D /D increases mono- becomes unstable if an axial uniform electric-current is strong tonically until a certain Re is reached, beyond it decreases. This enough. The rotation law between the two cylinders is ﬁxed by ∗ behavior can be understood by the fact that the ratio D /D their rotation; our main application is a quasi-Kepler rotation must be a direct function of the radial velocity magnitude and which results when the cylinders are rotating like planets. The ∗ the wave number of the solution, since D is produced solely magnetic Prandtl number of the ﬂuid is ﬁxed to the value of 0.1 by the instability which modiﬁes the radial velocity magnitude while its Schmidt number (12) is a free parameter of the model. and the wave number. As shown in Fig. 2 the magnitude of the The main result is a strictly linear relation between the result- radial velocity component hardly changes from Re = 400 to ing normalized eddy diffusivity D∗/D and the given Schmidt Re = 800 while the wavelength increases between Re = 400 number. The Schatzman relation (17) – which also describes our ∗ and Re = 700. Thus the decreasing of the ratio D /D is due to result that for small Schmidt number the eddy diffusivity is neg- the decreasing of the wave number. ligibly small – has thus been conﬁrmed in a self-consistent way. The model also provides numerical values for the scaling ∗ factor Re which increases for increasing Hartmann number of 16 the toroidal ﬁeld. For Schmidt numbers <∼ 0.1 the diffusivity due ∗ to the magnetic instability is negligibly small, however Re is 14 00 different from zero and even exceeds unity in our computations. 2 12 R e= 1 Already for the value Ha = 80 one ﬁnds Re∗ <∼ 4 for quasi- 1 80 H a= 200, = 1 2 0, R e = 8 0 0 KnuempTlbehererro(sFteaicgtoi.on6nd)a.rnedsuthltiscovnalcueerninsctrheeasreoslefoorfinthcreeamsainggneHtiacrtMmaancnh H a number Mm which represents the global rotation in relation to 46 H a = 8 0 , R e = 5 0 0 tihtye rmunasgnlientiecarﬁ(cid:39)ewldithstMrenmgtbhu.tFionr aslllowcasroestaatiomnatxhiemeudmdyodfiDffu∗s/iDv- exists at Mm 2. For faster rotation the induced diffusion is 2 suppressed and ﬁnally seems to remain constant (Fig. 5). This fast-rotation phenomenon may be a consequence of the Taylor- 0 Proudman theorem after which the axial ﬂuctuations are favored 0 0.5 1 1.5 2 in expense of the radial ones. Also the correlation lengths in ax- Sc ial direction grow for growing Reynolds numbers. Both conse- ∗ Fig. 6. D /D as a function of Sc for those Reynolds numbers quences of the Taylor-Proudman theorem appear to retard the yielding the maximal instability-induced diffusivities. The mag- growth of the radial diffusion in stars. netic Mach number slightly exceeds 2 in all cases. µ = 0.35, Figure 7 summarizes the results of this study by presenting Pm = 0.1. numerical values of Re∗ for Kepler rotation laws. In the upper ∗ panel Re is given for four values of the Hartmann numbers as a function of the Reynolds number. Without rotation one ﬁnds that Figure 6 shows the normalized diffusivity for µ = 0.35 ∗ ∝ for ﬁxed Ha as a function of Sc. For all Reynolds numbers the Re Ha is realized w∗ hich for solar/stellar Ha-values would induced diffusivities D∗ are different from zero and the ratio produce even higher Re -values that the one∗s found in our com- D∗/D scales linearly with Sc. The ﬁgure only shows the rela- putations. With rotation, a maximum of Re exists for app(cid:39)rox- tion for those Reynolds numbers for which the maximum dif- imately one and the sa∗me magnetic Mach number (Mm 2) fSucsi→viti0esthfeodrimffuasginveittyicDM∗aacphpneuarmsbtoervsaanbisohu.tH2eanrcee,refoarchmeodl.eFcuo-r wnuhmerbeert.heThvealhuieghoefrRthee Hstarortnmgalynninncurmeabseers twheithlatrhgeerHRaret∗m, abnunt this relation is not strictly linear as a mild saturation may exist. lar Schmidt numbers Sc > 0.1 the essence of Fig. 6 is the linear (cid:29) ∗ relation between D∗/D and Sc. In the notation of Schatzman For Mm 2 the rotational quenching leads to Re even smaller (1977) it means that than the values for Re = 0. ∗ ∗ The numerical restrictions of our code prevent calculations D = Re ν (17) for higher Reynolds numbers but we are able to vary the mag- ∗ with the scaling factor Re (which indeed forms some kind of a netic Prandtl number. This might be necessary as Pm in the solar Reynolds number). Also the Figs. 4 and 5 demonstrate that lin- tachocline is certainly smaller than 0.1. Figure 7 (middle panel) ∗ ear relations hold for all considered Re and Ha. The Schatzman shows the clear result that Re is anticorrelated with the mag- ∗ ∗ factor Re after Fig. 6 grows with growing Ha while for all three netic Prandtl number. The smaller Pm the larger Re , without ∗ ∝ models the magnetic Mach number is nearly the same. We ﬁnd rotation there is even Re 1/Pm. For fast rotation the results ∗ ∗ Re <∼ 4 for Ha = 80 growing to Re <∼ 8 for Ha = 200; a sat- do not exclude the possibility that a saturation may occur for 5 * D/ D ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A.

text: Paredes,

text: M.

text: Gellert,

text: G.

text: Ru¨diger:

text: Mixing

text: of

text: a

text: passive

text: scalar

text: Mach

text: number

text: becomes

text: unity

text: at

text: Re

text: In

text: the

text: slow

text: rota-

text: uration

text: for

text: larger

text: Ha

text: is

text: indicated

text: by

text: the

text: results

text: presented

text: in

text: Fig.

text: tion

text: regime,

text: the

text: effective

text: diffusivity

text: hardly

text: changes.

text: The

text: fact

text: It

text: is

text: not

text: yet

text: clear

text: whether

text: for

text: large

text: Ha

text: an

text: upper

text: limit

text: exists

text: that

text: even

text: without

text: rotation

text: the

text: ratio

text: D

text: /D

text: is

text: different

text: from

text: zero

text: for

text: Re

text: due

text: to

text: numerical

text: limitations.

text: is

text: because

text: the

text: ﬂow

text: is

text: unstable

text: even

text: for

text: Re

text: In

text: the

text: fast

text: rota-

text: tion

text: regime,

text: the

text: ratio

text: D

text: /D

text: increases

text: in

text: a

text: monotonic

text: way

text: until

text: Conclusions

text: it

text: peaks

text: at

text: about

text: Mm

text: Finally,

text: for

text: faster

text: rotation

text: (Mm

text: the

text: effective

text: diffusivity

text: decays

text: due

text: to

text: the

text: rotational

text: quenching.

text: The

text: inﬂuence

text: of

text: the

text: current-induced

text: instability

text: on

text: the

text: effec-

text: In

text: all

text: cases,

text: however,

text: the

text: effective

text: normalized

text: diffusivity

text: grows

text: tive

text: diffusivity

text: in

text: radial

text: direction

text: of

text: a

text: rotating

text: pinch

text: has

text: been

text: with

text: growing

text: Schmidt

text: number

text: so

text: that

text: also

text: here

text: simply

text: D

text: ν

text: studied

text: for

text: different

text: rotation

text: laws.

text: The

text: diffusion

text: equation

text: is

text: without

text: any

text: inﬂuence

text: of

text: the

text: microscopic

text: diffusivity.

text: The

text: miss-

text: numerically

text: solved

text: in

text: a

text: cylindric

text: setup

text: under

text: the

text: inﬂuence

text: of

text: ing

text: factor

text: in

text: this

text: relation

text: is

text: simply

text: given

text: by

text: the

text: curve

text: for

text: Sc

text: stochastic

text: ﬂuctuations

text: which

text: are

text: due

text: to

text: the

text: magnetic

text: Tayler

text: in-

text: in

text: Fig.

text: stability.

text: The

text: conducting

text: ﬂuid

text: between

text: two

text: rotating

text: cylinders

text: For

text: all

text: Schmidt

text: numbers,

text: the

text: ratio

text: D

text: /D

text: increases

text: mono-

text: becomes

text: unstable

text: if

text: an

text: axial

text: uniform

text: electric-current

text: is

text: strong

text: tonically

text: until

text: a

text: certain

text: Re

text: is

text: reached,

text: beyond

text: it

text: decreases.

text: This

text: enough.

text: The

text: rotation

text: law

text: between

text: the

text: two

text: cylinders

text: is

text: ﬁxed

text: by

text: behavior

text: can

text: be

text: understood

text: by

text: the

text: fact

text: that

text: the

text: ratio

text: D

text: /D

text: their

text: rotation;

text: our

text: main

text: application

text: is

text: a

text: quasi-Kepler

text: rotation

text: must

text: be

text: a

text: direct

text: function

text: of

text: the

text: radial

text: velocity

text: magnitude

text: and

text: which

text: results

text: when

text: the

text: cylinders

text: are

text: rotating

text: like

text: planets.

text: The

text: the

text: wave

text: number

text: of

text: the

text: solution,

text: since

text: D

text: is

text: produced

text: solely

text: magnetic

text: Prandtl

text: number

text: of

text: the

text: ﬂuid

text: is

text: ﬁxed

text: to

text: the

text: value

text: of

text: by

text: the

text: instability

text: which

text: modiﬁes

text: the

text: radial

text: velocity

text: magnitude

text: while

text: its

text: Schmidt

text: number

text: is

text: a

text: free

text: parameter

text: of

text: the

text: model.

text: and

text: the

text: wave

text: number.

text: As

text: shown

text: in

text: Fig.

text: the

text: magnitude

text: of

text: the

text: The

text: main

text: result

text: is

text: a

text: strictly

text: linear

text: relation

text: between

text: the

text: result-

text: radial

text: velocity

text: component

text: hardly

text: changes

text: from

text: Re

text: to

text: ing

text: normalized

text: eddy

text: diffusivity

text: D∗/D

text: and

text: the

text: given

text: Schmidt

text: Re

text: while

text: the

text: wavelength

text: increases

text: between

text: Re

text: number.

text: The

text: Schatzman

text: relation

text: which

text: also

text: describes

text: our

text: and

text: Re

text: Thus

text: the

text: decreasing

text: of

text: the

text: ratio

text: D

text: /D

text: is

text: due

text: to

text: result

text: that

text: for

text: small

text: Schmidt

text: number

text: the

text: eddy

text: diffusivity

text: is

text: neg-

text: the

text: decreasing

text: of

text: the

text: wave

text: number.

text: ligibly

text: small

text: has

text: thus

text: been

text: conﬁrmed

text: in

text: a

text: self-consistent

text: way.

text: The

text: model

text: also

text: provides

text: numerical

text: values

text: for

text: the

text: scaling

text: factor

text: Re

text: which

text: increases

text: for

text: increasing

text: Hartmann

text: number

text: of

text: the

text: toroidal

text: ﬁeld.

text: For

text: Schmidt

text: numbers

text: the

text: diffusivity

text: due

text: to

text: the

text: magnetic

text: instability

text: is

text: negligibly

text: small,

text: however

text: Re

text: is

text: different

text: from

text: zero

text: and

text: even

text: exceeds

text: unity

text: in

text: our

text: computations.

text: R

text: e=

text: Already

text: for

text: the

text: value

text: Ha

text: one

text: ﬁnds

text: Re∗

text: for

text: quasi-

text: H

text: a=

text: R

text: e

text: H

text: a

text: number

text: Mm

text: which

text: represents

text: the

text: global

text: rotation

text: in

text: relation

text: to

text: H

text: a

text: R

text: e

text: tihtye

text: exists

text: at

text: Mm

text: For

text: faster

text: rotation

text: the

text: induced

text: diffusion

text: is

text: suppressed

text: and

text: ﬁnally

text: seems

text: to

text: remain

text: constant

text: (Fig.

text: This

text: fast-rotation

text: phenomenon

text: may

text: be

text: a

text: consequence

text: of

text: the

text: Taylor-

text: Proudman

text: theorem

text: after

text: which

text: the

text: axial

text: ﬂuctuations

text: are

text: favored

text: in

text: expense

text: of

text: the

text: radial

text: ones.

text: Also

text: the

text: correlation

text: lengths

text: in

text: ax-

text: Sc

text: ial

text: direction

text: grow

text: for

text: growing

text: Reynolds

text: numbers.

text: Both

text: conse-

text: Fig.

text: D

text: /D

text: as

text: a

text: function

text: of

text: Sc

text: for

text: those

text: Reynolds

text: numbers

text: quences

text: of

text: the

text: Taylor-Proudman

text: theorem

text: appear

text: to

text: retard

text: the

text: yielding

text: the

text: maximal

text: instability-induced

text: diffusivities.

text: The

text: mag-

text: growth

text: of

text: the

text: radial

text: diffusion

text: in

text: stars.

text: netic

text: Mach

text: number

text: slightly

text: exceeds

text: in

text: all

text: cases.

text: µ

text: Figure

text: summarizes

text: the

text: results

text: of

text: this

text: study

text: by

text: presenting

text: Pm

text: numerical

text: values

text: of

text: Re∗

text: for

text: Kepler

text: rotation

text: laws.

text: In

text: the

text: upper

text: panel

text: Re

text: is

text: given

text: for

text: four

text: values

text: of

text: the

text: Hartmann

text: numbers

text: as

text: a

text: function

text: of

text: the

text: Reynolds

text: number.

text: Without

text: rotation

text: one

text: ﬁnds

text: that

text: Figure

text: shows

text: the

text: normalized

text: diffusivity

text: for

text: µ

text: for

text: ﬁxed

text: Ha

text: as

text: a

text: function

text: of

text: Sc.

text: For

text: all

text: Reynolds

text: numbers

text: the

text: Re

text: Ha

text: is

text: realized

text: w∗

text: hich

text: for

text: solar/stellar

text: Ha-values

text: would

text: induced

text: diffusivities

text: D∗

text: are

text: different

text: from

text: zero

text: and

text: the

text: ratio

text: produce

text: even

text: higher

text: Re

text: -values

text: that

text: the

text: one∗s

text: found

text: in

text: our

text: com-

text: D∗/D

text: scales

text: linearly

text: with

text: Sc.

text: The

text: ﬁgure

text: only

text: shows

text: the

text: rela-

text: putations.

text: With

text: rotation,

text: a

text: maximum

text: of

text: Re

text: exists

text: for

text: tion

text: for

text: those

text: Reynolds

text: numbers

text: for

text: which

text: the

text: maximum

text: dif-

text: imately

text: one

text: and

text: the

text: sa∗me

text: magnetic

text: Mach

text: number

text: (Mm

text: wnuhmerbeert.heThvealhuieghoefrRthee

text: Hstarortnmgalynninncurmeabseers

text: twheithlatrhgeerHRaret∗m,

text: abnunt

text: this

text: relation

text: is

text: not

text: strictly

text: linear

text: as

text: a

text: mild

text: saturation

text: may

text: exist.

text: lar

text: Schmidt

text: numbers

text: Sc

text: the

text: essence

text: of

text: Fig.

text: is

text: the

text: linear

text: relation

text: between

text: D∗/D

text: and

text: Sc.

text: In

text: the

text: notation

text: of

text: Schatzman

text: For

text: Mm

text: the

text: rotational

text: quenching

text: leads

text: to

text: Re

text: even

text: smaller

text: it

text: means

text: that

text: than

text: the

text: values

text: for

text: Re

text: The

text: numerical

text: restrictions

text: of

text: our

text: code

text: prevent

text: calculations

text: D

text: Re

text: ν

text: for

text: higher

text: Reynolds

text: numbers

text: but

text: we

text: are

text: able

text: to

text: vary

text: the

text: mag-

text: with

text: the

text: scaling

text: factor

text: Re

text: (which

text: indeed

text: forms

text: some

text: kind

text: of

text: a

text: netic

text: Prandtl

text: number.

text: This

text: might

text: be

text: necessary

text: as

text: Pm

text: in

text: the

text: solar

text: Reynolds

text: number).

text: Also

text: the

text: Figs.

text: and

text: demonstrate

text: that

text: lin-

text: tachocline

text: is

text: certainly

text: smaller

text: than

text: Figure

text: (middle

text: panel)

text: ear

text: relations

text: hold

text: for

text: all

text: considered

text: Re

text: and

text: Ha.

text: The

text: Schatzman

text: shows

text: the

text: clear

text: result

text: that

text: Re

text: is

text: anticorrelated

text: with

text: the

text: mag-

text: factor

text: Re

text: after

text: Fig.

text: grows

text: with

text: growing

text: Ha

text: while

text: for

text: all

text: three

text: netic

text: Prandtl

text: number.

text: The

text: smaller

text: Pm

text: the

text: larger

text: Re

text: without

text: models

text: the

text: magnetic

text: Mach

text: number

text: is

text: nearly

text: the

text: same.

text: We

text: ﬁnd

text: rotation

text: there

text: is

text: even

text: Re

text: For

text: fast

text: rotation

text: the

text: results

text: Re

text: for

text: Ha

text: growing

text: to

text: Re

text: for

text: Ha

text: a

text: sat-

text: do

text: not

text: exclude

text: the

text: possibility

text: that

text: a

text: saturation

text: may

text: occur

text: for

text: D/

text: D

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##