On the relevance of bubbles and potential flows for stellar convection

2.1 MB·

by  Marcelo M. Miller Bertolami

MNRAS000,1–14(2015) Preprint25January2016 CompiledusingMNRASLATEXstylefilev3.0
On the relevance of bubbles and potential flows for stellar
convection
M. M. Miller Bertolami,1,2⋆ M. Viallet,1 V. Prat1, W. Barsukow3 and A. Weiss1
1Max-Planck-Institut fu¨rAstrophysik, Karl-Schwarzschild-Str. 1, 85748, Garching, Germany.
2Institutode Astrof´ısica de La Plata, UNLP-CONICET, Paseo delBosque s/n, 1900 La Plata, Argentina.
3Institut fu¨r Mathematik, Universit¨atWu¨rzburg, Campus Hubland Nord, Emil-Fischer-Strasse 40, 97074 Wu¨rzburg, Germany.
6
1
0
2 Accepted2016January20.Received2016January20;inoriginalform2015November06

n
a
ABSTRACT
J
Recently Pasetto et al. have proposed a new method to derive a convection theory
1
appropriatefortheimplementationinstellarevolutioncodes.Theirapproachisbased
2
on the simple physical picture of spherical bubbles moving within a potential flow

in dynamically unstable regions,and a detailed computation of the bubble dynamics.
]
R Basedonthisapproachtheauthorsderiveanewtheoryofconvectionwhichisclaimed
S tobeparameterfree,non-localandtime-dependent.Thisisaverystrongclaim,assuch
. a theory is the holy grail of stellar physics. Unfortunately we have identified several
h
distinct problems in the derivation which ultimately render their theory inapplicable
p
to any physical regime. In addition we show that the framework of spherical bubbles
-
o in potential flows is unable to capture the essence of stellar convection, even when
r equations are derived correctly.
t
s
a Key words: convection – stars: fundamental parameters – stars: evolution
[

1
v
1 1 INTRODUCTION (Xiong 1986; Weiss & Flaskamp 2007). Attempts to derive
1 a general framework for the treatment of stellar convec-
8 It is not an exaggeration to state that the turbulent trans- tion andothermixingprocesses lead toverycomplex equa-
5 portofheat,angularmomentumandchemicalspeciesisthe
tionswhichcannotbeeasilyincludedin1Dstellarevolution
0 most important unsolved problem in stellar astrophysics.
codes (Kuhfuss 1986; Canuto 2011a,b,c,d,e). In fact, non-
. Most of the present uncertainties in stellar physics are, in
1 localconsequencesofconvection,suchasconvectivebound-
one way or another, linked to our incomplete understand-
0 arymixing,areroutinelyincludedinstellarevolutioncodes
6 ing of mixing in stellar interiors, e.g. the final fate of stars based on ad-hoc prescriptions and additional free parame-
1 of high and intermediate mass, formation of s-process el-
ters —see Viallet et al. (2015) and Arnettet al. (2015) for
: ements, chemical anomalies on the red giant branch, for-
v recent discussions on these issues. Consequently,despite its
mation of carbon stars, size of the convective cores in H-
i well-known shortcomings, the mixing-length theory (MLT;
X andHe-burningstars. Inspiteof manydecadesof attempts
Prandtl1925;Biermann1932;Vitense1953)hasbeeninuse
r to derive an accurate time-dependent and non-local the- for more than 80 years.
a ory of convection that can be included in stellar evolution
codes, success has been very minor. While some theories of
time-dependent convection have been derived and applied In this paper, we call a “theory of stellar convection”
(Kuhfuss 1986; Kuhfuss, R. 1987; Wuchterl& Feuchtinger a theory that can be implemented in 1D stellar evolution
1998; Flaskamp, M. 2003), they all introduce several free codes.Suchatheoryshouldcapturetheessentialproperties
parameters that must be calibrated for different regimes, of turbulentconvection,allowing to reproduceits effectson
diminishing their predictive power. Even more problematic the stellar structure (mainly chemical mixing and energy
isthecaseofnon-localconvectionandconvectiveboundary transport)withouttheneedtoresorttoexpensive3Dsimu-
mixing.Fordecades,seriousattemptshavebeenmadetode- lations.Atheoryofstellarconvectionishighlysought,asthe
rive non-local convection theories that could be introduced predictivepowerofcurrentstellarmodelsisstronglylimited
in stellar evolution codes (Denget al. 2006; Deng& Xiong by the shortcomings of MLT. The status on the theoreti-
2008; Canuto 2011e). However, these theories are not pop- cal side contrasts with the progress done in observational
ular due to their complexity and their limited accuracy techniquesandinstrumentation(e.g.KEPLER,CoRoTand
GAIA; deBruijne 2012; Guzik et al. 2014). A new genera-
tion of stellar models is necessary to fully exploit the large
⋆ E-mail:[email protected] amountofqualitydatathatisdeliveredbyobservers.With-
c 2015TheAuthors
(cid:13)
2 M. M. Miller Bertolami et al.
outanydoubtanewgenerationofstellarmodelsshouldrely assess the physical regime in which the method described
on a bettertreatment of convection. by Pasetto et al. (2014) can be applied. Unfortunately we
Recently, Pasetto et al. (2014) claimed to present an find that the movement of spherical bubbles within poten-
accurate parameter-free, non-local, time-dependent theory tial flows is completely inapplicable to the regime of stellar
of stellar convection that can be easily implemented in convection and that no useful theory of stellar convection
1D stellar codes. This is a strong claim since such a the- can be obtained from this approach. This is not a surprise
ory has been sought for many decades. In order to facili- since the adoption of the ideal fluid and the potential flow
tate the reader’s understanding, we start by schematically approximations (the“dry water” approximation, Feynman
summarizingthemethodproposedbyPasetto et al.(2014). 1964)neglecttheimportanceofviscosityandboundarylay-
Pasetto et al.(2014)adoptarathersimplepictureofconvec- ers for thedynamicsof thebubble.
tion,inwhichthetransportofheatisachievedby“bubbles” The paper is organized as follows. In the next section
thatriseduetobuoyancyin aconvectivelyunstableregion. we show that the derivation of Pasetto et al. (2014) of the
This description of convection using theconcept of bubbles accelerationequationisflawedduetoincorrectphysicaland
islikely inspiredfrom theusualsimplepicturethat onehas mathematical assumptions. In section 3 we clarify the ap-
inmindwhenderivingtheMLT.Furthermore,inthepicture proximations underlyingtheir theory of an isolated bubble,
of Pasetto et al. (2014), convective bubbles have a definite and providethecorrect derivation of theacceleration equa-
shape (they are spherical) and are differentiated from the tionofthebubble.Weshowthattheauthorsmisinterpreted
surrounding material —i.e. the surrounding material flows their acceleration equation, and neglected a term that is
aroundthem.Asafirststeptheauthorsanalyzethemotion physically important. In section 4 we provide the correct
of an isolated bubble. From kinematic considerations they analysis of the equation of motion, and focus particularly
derivetheexpressionofthevelocityfieldaroundthebubble. on the asymptotic/finalregimes reached bythe bubble.We
This is done assuming that the flow around thebubbleis a show that it is unavoidable that the theory becomes incon-
potential flow ( u = 0, where u is the velocity field). sistent and highly non-physical. In section 5, we finish the
∇×
This allows them to link the velocity of the moving bubble article with some discussion and concluding remarks.
to that of the surrounding fluid at each time —i.e. assum-
ing an instantaneous adjustment of the surrounding fluid.
Giventhevelocityfieldaroundthebubble,theauthorsthen
2 PHYSICAL AND MATHEMATICAL
deduce the pressure field around the bubble. Knowing the
INCONSISTENCIES IN PASETTO ET AL.
pressure field, they compute the total force that the fluid
(2014)
exertson thebubble,
Before analyzing the physical and mathematical assump-
F~ = P~ndS. tions adopted by Pasetto et al. (2014), it is already worth
I
bubble noting that a first consequence of the method adopted by
Applying Newton’s law to the bubble, the authors derive the authors is that it cannot provide a self-consistent time-
an expression for the acceleration of the bubble, the first dependent and non-local convection theory in the usual
key result of their theory. With appropriate initial condi- sense of these terms. By looking at the system of equa-
tions, this equation defines completely the motion of the tions that define the theory of convection presented in
bubble as a function of time. In the second part of their Pasetto et al. (2014), see their eqs. [60]1, it becomes appar-
work, Pasetto et al. (2014) use their theory for an isolated entthattheirtheoryisalocalformulation,verymuchinthe
bubbletoformulateatheoryofconvectionbyconsideringa spiritofMLT.Inalocaltheoryofconvection,velocitiesand
collective set of bubbles. convectivefluxesdependonlyonthelocalthermodynamical
In this work we study the applicability of this method variablesandtheirlocalgradients.Usually,alocaltheoryof
to the stellar regime and its possible limitations. In order convectionresultsfroma“local”approachtotheproblemof
to do this we derive the equation for the dynamics of the convection. In a local approach, one makes the assumption
bubble by a careful accounting of the physical assumptions that all the relevant processes are taking place on length-
and hypothesis made in the derivation. During this process scales l that are much smaller than the typical length scale
we found that some inconsistent physical and mathemati- over which the background is changing, i.e. l Hp,Hρ,
≪
cal assumptions have been made by Pasetto et al. (2014), wherel isthelengthscaleoftheprocessofinterest,Hp and
castingseriousdoubtsonthevalidityoftheirtheory.Itwill Hρ are the pressure and density scale height, respectively.
also become clear in the next section that the claim of a Clearly, the work presented in Pasetto et al. (2014) follows
non-local,time-dependenttheoryisanoverstatementbythe such a local approach, as clearly stated in their Sect. 22. It
authors. Yet, the method of deriving a parameter-free con-
vection theory from the full dynamics of a convective ele-
ment assuming a surrounding potential flow is interesting. 1 Throughout this paper we denote the equations in
If valid, the method could indeed be extended to obtain a Pasettoetal. (2014) with square brackets to differentiate
parameter-free,non-localandtime-dependenttheoryandto themfromourownequations.
2 Wheretheystate,“Weproceed furtherwithanadditional sim-
getridofthemixinglengthparameterαwhosecalibrationin
plification by assuming that the stellar fluid is incompressible
different stellar regimes is problematic (Ludwig et al. 1999;
and irrotational on large distance scales. The concept of a large
Trampedach et al. 2014; Tremblay et al. 2015; Magic et al.
distance scale for incompressibility and irrotationality is defined
2015). We have been able to reobtain the dynamics of the here from a heuristic point of view: This length should be large
bubble by a sound mathematical and physical derivation. enoughtocontainasignificantnumberofconvectiveelementsso
This allow us to study the behavior of the solutions and thatastatisticalformulationispossiblewhendescribingthemean
MNRAS000,1–14(2015)
Bubbles, potential flows and stellar convection. 3
is not possible to derive a self-consistent non-local theory thanitsexpansionvelocityR˙ = R˙ —throughoutthiswork
| |
of convection from such a local approach, as it is precisely we denote the radius of the bubble by R, and its temporal
the local approach that decouples the problem at each ra- changesbyR˙ andR¨.However,itiseasytoshowthatsucha
dius.Furthermore,a“time-dependent”theoryof convection regime is in strong contradiction with the assumptions of a
has a very specific meaning in the field of 1D stellar struc- subsonicregime andalocal approach—thelattermaterial-
turecomputations.Itreferstoatheorywhich isabletode- izedbythepossibilityofsolvingthemovementofthebubble
scribe convection in the case where the stellar background assuming a medium of constant density.
evolves on a timescale smaller, or of the same order, than Let us say that a bubble is characterized by its mass
the convectiveturn-overtimescale. As mentioned in the in- m (constant in time), density ρ (t), pressure P (t), radius
b b b
troduction,such theories exist but theirpredictivepower is R(t), position r (t), and velocity v (t) =r˙ . The surround-
b b b
hampered by several free parameters. As admitted by the ing medium is characterized by its pressure stratification
authors in one of their footnotes, the theory presented in P(r).First, a spherical bubbletraveling in the surrounding
Pasetto et al. (2014) is not “time-dependent”in the usual mediumatasubsonicspeedremainsinpressureequilibrium,
sense3. Very likely, Pasetto et al. (2014) refer to their the- i.e.P P assoundwavesareabletowashoutanypressure
b
oryasbeing“time-dependent”becausetheyintegrateintime differen≃ce4.ThereforeP (t)=P(r (t))and,takingthetime
b b
a set of equations until an asymptotic regime is obtained. derivative,oneobtains
However, their theory of convection is based on theasymp-
totic regime, where the time variable is not relevant any dPb(t) = dP(rb(t)) = dP vb = Pvb, (2)
moreand,consequentlyit cannotbeconsidered as atheory dt dt dr −HP
of time-dependentconvection. or simply
Having clarified that the approach derived by dlogP v
Pasetto et al. (2014) deals with a time-independentand lo- b = b . (3)
dt −HP
caltheorywenowturntoanalyzesomeofthemathematical
andphysicalapproximationsmadeintheirderivationofthe We used the definition of the pressure scale-height HP =
dr . Neglecting heat conduction, the change in density
equation of motion for thespherical bubble. −dlogP
of thebubblefollows theadiabatic relation
P
b =const, (4)
2.1 The physical assumptions ρbΓ1
After deriving the equations for the velocity field of an where Γ1 is thefirst adiabatic index.This is equivalentto
incompressible and irrotational fluid around an expanding
dlogP dlogρ
spheremoving within a fluidof constant density and in hy- b =Γ1 b. (5)
dt dt
drostaticequilibriumatinfinity(seetheirsections2and3),
Combining eqs.3 and 5,we obtain:
the authors apply this result to compute the forces exerted
onthespherebythesurroundingfluid.Besidestheassump- dlogρ v
tions of an incompressible and irrotational fluid of constant Γ1 dt b =−HbP. (6)
density, they also neglect heat diffusion and restrict them-
Finally, as the mass of the bubble is constant, its density
selves tothesubsonicregime (i.e. spheresmovingat speeds decreases as ρ R−3. Thus,we obtain:
b
muchsmallerthanthespeedofsound).Inthiscontextthey ∝
claimthatitisreasonabletoassumethat(seetheireq.[12]) 3Γ1dlogR = vb . (7)
v − dt −HP
R˙b ≪1, ∀t>tmin, (1) It follows that, within the adiabatic and subsonic approx-
imations, the relation between the expansion rate and the
i.e. that the relative velocity vb =|vb| between the convec- velocity of the bubbleis
tive element and the intrastellar medium is much smaller
vb = 3HPΓ1. (8)
R˙ R
convectivefluxofenergy(seebelow),butsmallenoughsothatthe
distance traveled by the convective element is short compared to Wecanconcludethattheassumptionvb/R˙ 1isequivalent
≪
the typical distance over which significant gradients in tempera- toHP/R 1, as usually Γ1 1.
≪ ∼
ture, density, pressure, etc. can develop (i.e. those gradients are This result can be understood on the basis of the fol-
locally small).” lowingverysimplephysicalobservation.Withinthesubsonic
3 This ishinted by the authors at the end of section 2, p. 3594; approximation,theonly way in which a bubblecan expand
“Beforestartingouranalysis,inordertoavoidapossiblemisun-
much faster than it moves is when small vertical displace-
derstanding of the real meaning of some of our analytical re-
mentsleadtobigchangesinthepressureofthesurrounding
sults, it might be wise to call attention to a formal aspect of
the mathematical notation we have adopted. For some quanti- fluid,i.e.whenHP isverysmallcomparedtothesizeofthe
ties Q function of time or space or both, Q(x;t), we look at bubble.
their asymptotic behaviour by formally taking the limits Q∞ = Unfortunately, assuming vb/R˙ 1, which implies
≪
limx→x∞,t→∞Q(x;t). This does not mean that we are taking HP/R ≪ 1, is in complete contradiction with the core of
temporal intervals infinitely long, rather that we are considering
time long enough so that the asymptotic trend of the quantity Q
isreachedbutstillshortenoughsothatthephysicalpropertiesof 4 In addition, we show in appendix A that this is also mathe-
thewhole systemhavenotchangedsignificantly,suchasthatthe matically consistent with the equations for the dynamics of the
star stillexists.”. bubbletobederivedlater.
MNRAS000,1–14(2015)
4 M. M. Miller Bertolami et al.
the theory which is based on a local picture of convection. second term becomes negligible is the one of strong buoy-
In particular it is in clear contradiction with expressions ancy forces (M m )/m 1. A simple rewriting of their
b b
suchaseqs.[3],[13],[24]and[27]fromPasetto et al.(2014) eq.[24] using th−edefinition∼of M =4πR3ρ/3, shows that
which are derived within the picture of a bubblemoving in
a constant density background. v˙b=g mb−M 10Mvb(R˙/R). (12)
mb+M/2 − 4 mb+M/2
It follows that, for strong buoyancyforces, thesecond term
2.2 The mathematical approximations becomes negligible when g vbR˙/R. Using that HP =
Whilethepreviousinconsistencyisseriousenoughtorender P/gρ and that for an ideal ≫gas the sound speed is cs2 =
γP/ρ, we see that the second term becomes negligible if
theapplicabilityofthetheoryquestionable,othercontradic-
tions develop as a consequence of mathematical simplifica- cs2/γHp ≫ R˙/Rvb. As the derivation of the equation of
toivoenrstdhuerminogvitnhgesdpehreivreat—ioSneocftitohnesf4o.2rcaenedx5erotefdPabsyettthoeefltuaild. cmoontdioitniownithhoilndsaaloscsaolopnicatsurces2r/eqγu≫ireRs˙HvbP.A≫saR,retshueltp,rweveiosuees
thatthesecondtermisindeednegligibleassoonaswehave
(2014). The first of these approximations comes during the
significantbuoyancyforces(M m )/m &1andwerestrain
derivation of“Lemma 1”of Pasetto et al. (2014) (eq. [13]). − b b
Thereitisstated that,underthevalidityofv /R˙ 1,it is ourselvestosubsonicmotionsandexpansions.Theprevious
b
≪ argumentshowsthat,althoughforverydifferentreasons,in
possible to say that
theregimeofsignificantbuoyancyandsubsonicbubblesthe
v 2 1 9 R 3 R¨R keyequation [26] of Pasetto et al. (2014) is valid.
(cid:18)R˙b(cid:19) 2(cid:18)4sin2θ−1(cid:19)≪v˙bR˙2 (cid:18)2cosθ−cosφ(cid:19)+ R˙2 , Finally,aseriousinconsistencyarisesduringtheircom-
putationoftheconvectivefluxintheirsection6.Inorderto
(9)
compute the velocity of the convective elements (their eq.
and also that [41]) the authors analyze the movement of the stagnation
pointsinthecase of anon-expandingrigid-bodymovement
vbR˙ 2 5cosθ v˙ R 3cosθ cosφ + R¨R. (10) (R˙ = R¨ = 0). The approximation of a non-expanding con-
(cid:18) R˙2 (cid:19) 2 ≪ bR˙2 (cid:18)2 − (cid:19) R˙2 vective element is in stark contradiction with the previous
It is clear that it is not possible to justify these two in- derivation of theory. Furthermore, the authors wrongly as-
equalities(eqs.[14]inPasetto et al.2014)solelyonthebase sume that P/ρ+Φg 0 at the stagnation points. From
≃
of v /R˙ 1 without any other assumption. In order to this analysis, Pasetto et al. (2014) conclude that the veloc-
b
justify eq≪s. 9 and 10 one must make the assumptions that ity, radius and acceleration of the bubble are connected by
v2 v˙ R and v2 R¨R. These two assumptions restrict (see theireq.[41])
b ≪| b | b ≪| |
evenmorethephysicalregimeinwhichthetheorycouldbe v2 = v˙ R. (13)
applicable.Onemightwonderwhethersuchspecificregime, b − b
i.e. v /R˙ 1, v2 v˙ R and v2 R¨R,does exist at all. Clearly, assuming eq. 13 is in apparent contradiction with
Wb e w≪ill shobw≪la|tebr t|hat theb ≪two| inc|orrect approxima- eqs. 9 and 10, which require vb2 ≪ |v˙bR|. The neglection of
tions performed in eqs. 9 and 10 do not change the shape the second term in eq. 12 is also in contradiction with the
othfotuhgeheqtuhaetyiodnofocrhathnegeacscoemleeraotifonthoefctoheeffiflcuieidntesl.emUennfot,rtaul-- saismsuumltpantieoonussiamsspulympvtbiRo˙n/Rofvb2 =v˙b.−Wv˙beRwainlldsehqo.w1—inassetchtieosne
nately,afterthederivationtheequationofmotion(theireq. 4 that the ratio vb2/(v˙bR) c≫hanges by orders of magnitude
[24])5 during the actual motion of the bubble(see Figs. 1 and 2).
Consequentlyeq. 13 does not hold.
v˙b=g mb−M 10 πR2ρvbR˙ , (11)
mb+M/2 − 3 mb+M/2
the authors simplify this expression by neglecting the sec- 3 EQUATION OF MOTION FOR AN
ond term to obtain their eq. [26]. It is not possible to ne- EXPANDING SPHERE IN A POTENTIAL
glect the second term solely on the base of v /R˙ 1 as FLOW
b
≪
it is claimed by Pasetto et al. (2014). The physical regime
As mentioned during the introduction, during the study of
in which this term can be neglected is discussed below. It
Pasetto et al. (2014) we found that the equivalent of their
is worth noting that their eq. [26] plays a key role in the
keyequation[24](eq.11)canbederivedinasoundphysical
derivation of the convective theory, as it is eq. [26] that is
and mathematical way. This is an interesting result which
used in the further development of the work —e.g. in the
willallowustostudythemotionofanisolatedbubblewithin
derivation of their eq. [27]. Interestingly, by doing this the
the present picture and assess its applicability to derive a
authorsdroppedtheonlytermthatcouldprovidethemwith
theory of stellar convection.
a truly asymptotic regime, as we will show in section 4. It
In line with Pasetto et al. (2014) we will assume that
is easy to see that, the actual physical regime in which the
the fluid is ideal (no viscosity), incompressible ( v = 0)
∇·
andirrotational ( v=0).Wewill assumethat thepath
∇×
traveledbythesphere(l )canbeconsideredsmallcompared
5 We have corrected the sign of the first term, because when b
tothedistancesoverwhichpressureP,gravitygordensity
M >mb(morebuoyancythanweight)thedirectionofv˙bshould
beoppositetothatofgand,also,haveaddedthedenominatorof ρ change. If HP and Hρ are the pressure and density scale
thefirstterm(mb+M/2)whichshouldalsoappearinthesecond heights we have lb ≪ HP and lb ≪ Hρ. The medium is as-
term. sumedtobeinhydrostaticequilibriumfarfrom themoving
MNRAS000,1–14(2015)
Bubbles, potential flows and stellar convection. 5
element ( P∞ =ρg; where P∞ means the pressure in that (x rb(t))/x rb(t) are functions of x and t. One can
∇ − | − |
layer and far away from thebubble). show that this velocity fields satisfies eqs. 14, 15 and 17.
This can be easily shown by noting that x′ = x rb(t)
−
implies that for any function F(x), ′F(x) = F(x), t.
3.1 Flow around an expanding sphere moving at ∇ ∇ ∀
The potential ψ that produces the field v (eq. 23) is given
constant velocity
by
Uthnadter tψhe=asvsu.mTphteiopnot∇en×tiavl=of0anthienrceoimsparepsostibenletiafllowψ soof ψ(x)= 1R3vb n′ R˙R2. (24)
consta∇nt density must fulfill 2ψ = 0 — see section 9 of −2r′2 · − r′
∇
Landau & Lifshitz(1987)foradetaileddiscussionofpoten-
tial flows. In particular the solution corresponding to the 3.2 The instantaneous adjustment hypothesis
motion (with velocity vb = vbez) of an expanding sphere In the following, we will assume that the shape of the ve-
(of radius R and expansion rate R˙) within a fluid which is
locity field instantaneously adjusts itself to the shape pre-
in hydrostatic equilibrium far away (i.e. x ) can be
| | → ∞ scribedbyeq.23fortheinstantaneousvaluesofvb(t),R(t)
obtained by solving and R˙(t), i.e. we assume that
2ψ=0, (14)
∇ 3R(t)3 R˙(t)R(t)2 1R(t)3
t, v(x,t)= n′(v n′)+n′ v ,
with theboundaryconditions ∀ 2r′3 b· r′2 − 2 r′3 b
(25)
t, lim v = 0, (15)
∀ |x|→∞ wheretheposition of thebubbleis given byrb(t) and n′ =
∀t,∀n′,v·n′ = R˙ +vb·n′, (16) x′/x′ ,wherex′=x rb(t)isthepositionasseenfromthe
| | −
on thesphere x rb =R, center of the bubble. The velocity field of eq. 25 fulfills the
| − |
boundary conditions given by eqs. 15 and 17 at every time
wherewedenotethepositionofthebubblebyrb(t)andwe
t. As t and x are independent variables, it is easy to show
define n′ = x′/x′, with x′ = x rb the position as seen
| | − that thepotential ψ(x,t) that produces thisfield is
from thecenter of thebubble.
Itiseasiertosolvetheproblembychangingtothecoor- 1R(t)3 R˙(t)R(t)2
dinate system comoving with the sphere at constant veloc- ψ(x,t)=−2 r′2 vb·n′− r′ . (26)
ityvb.Fromthatcoordinatesystemtheproblemreducesto In order for this hypothesis to hold, the fluid needs to
thatofanexpandingsphereatrestlocatedatx′=0within
adjustfast enoughtotheinstantaneousvelocityofthebub-
a fluidmoving at infinitywith v∞ =−vb, i.e. tosolving ble.Thishypothesiswillholdifboththeexpansionvelocity
′2ϕ=0, (17) of the sphere and the translational velocity of the sphere
∇ are much smaller than the sound speed, i.e. if vb cs and
wthheerbeou∇n′dadreynoctoensdtithioendserivatives with respect to x′, with lRa˙t≪edctso. Itnheadacdciteiloenrawtioenalasondastshuemcehtahnagtetihnetthimeee≪sxcpaalenssiroen-
lim v′ = v , (18) rate are small compared with the reaction timescale of the
r′→∞ ∞ fluid given by τ = R/cs— i.e. we assume that changes in
n′,v′ n′ = R˙, at x′ =R, (19) vb and R˙ fulfill R˙/R¨ R/cs and vb/v˙b R/cs. Underthe
∀ · | | assumption of subson≪icflows, this implie≪s that v˙ c2/R
wthheerceomvo′v=ing∇s′yϕstedmen,oatnesdtrh′e=velxo′ci.tyItfiiesldstraasigshetefnorwfraormd andR¨ ≪c2s/R.Notethattheassumptionofsub|sobn|i≪cvesloc-
| | ities is also compatible with the incompressibility approxi-
to checkthat the solution tothat problem is given by
mation, which implies cs = .
∞
1R3 R˙R2
ϕ(x′)= v n′ +v r′. (20)
2r′2 ∞· − r′ ∞·
3.3 Equation of motion for a moving and
This is an extension of the solutions discussed in sections expanding sphere within a fluid at rest
10 and 11 of Landau & Lifshitz (1987) in the case of an
expandingsphere. Computing thederivativeswe get Once the velocity field is known, one can use this result to
computetheforceexertedbythefluidonthemovingbubble
v′ = 3R3n′(v n′)+n′R˙R2 + 1R3v +v . (21) byusing Euler’s equation
−2r′3 ∞· r′2 2r′3 ∞ ∞
P
Thevelocity fieldasseen from thesystem inwhich the ∂tv+(v )v= ∇ +g, (27)
·∇ − ρ
bubble is in movement with velocity vb can be obtained
from a direct galilean transformation: where g = −∇Φg is the gravitational acceleration. For an
incompressibleandirrotationalfluidofconstantdensity,eq.
v(x)=v′(x′)+vb=v′(x rb)+vb. (22) 27 bewritten as
−
Usingv∞ =−vb we find, ∂tv+ v2 = P Φg. (28)
∇ 2 −∇ρ −∇
3R3 R˙R2 1R3 Eq.28 can then berewritten, using ψ=v, as
v= n′(v n′)+n′ v , (23) ∇
2r′3 b· r′2 − 2r′3 b v2 P
where it is worth noting that r′ = x rb(t) and n′ = ∇(cid:18)∂tψ+ 2 + ρ +Φg(cid:19)=0. (29)
| − |
MNRAS000,1–14(2015)
6 M. M. Miller Bertolami et al.
Integratingthis equation in space we find (38)
v2 P Integratingoverthe whole spherewe get
∂tψ+ + +Φg+c(t)=0, (30)
2 ρ
v2 4π
where c(t) is a constant of integration. It can be obtained n′dS = R˙R2vb, (39)
Z 2 3
∂V
by noting that for x the fluid is static (v = 0) and
in hydrostaticequil|ib|ri→um∞( (P/ρ+Φg)=0).This implies where we haveused that ez =ez′ and vb =vbez.
that6 ∇ Finally, the last integral in the RHS of eq. 34 can be
integrated using that
P
+Φg =C′, (31)
(cid:18)ρ (cid:19)|x|→∞ Φgn′dS = ΦgdV = gV(t), (40)
Z Z ∇ −
whereC′ isaconstantthatdependsonthearbitrarychoice ∂V V
of the definition of the gravitational potential. Noting that where V(t) = 4πR(t)3/3 is the volume of the expanding
for x we havethat ∂tψ 0 andv2 0, we see that sphere.
eq.|30|i→mp∞liesthatc(t)= C′.→Forthesake→ofsimplicitywe Usingeqs.36,38,and40ineq.34,theforceexertedby
−
can set C′ =c(t)=0, and we obtain thefluidon themovingbubbleis
P v2 4πR3 2π 2π
ρ =−∂tψ− 2 −Φg. (32) F=−Z Pn′dS =− 3 ρg− 3 ρR2R˙vb− 3 ρR3v˙b.
∂V
The force F applied to the bubble is obtained by inte- (41)
grating eq. 32 overthesurface of thesphere ∂V(t),
F= Pn′dS (33) 3.4 The acceleration of the bubble
−Z
∂V
v2 Theequationofmotionforthemovingsphere,underallthe
=ρ ∂tψn′dS+ρ n′dS+ρ Φgn′dS. (34) previously mentioned assumptions, is
Z Z 2 Z
∂V ∂V ∂V
The first integral in theRHS of eq.34 can beobtained mbv˙b= Pn′dS+mbg, (42)
usingthedefinitionofψ,takingthetimederivative∂tψand −Z∂V
evaluatingover thesphere. Wehave where m is the mass of the bubble (m =4πR3ρ /3), and
b b b
3 R the pressure integral is given by 41. Using the definition
∂tψ= R˙(vbn′) (v˙bn′) R¨R 2R˙2, for x rb(t) =R. M = 4πR3ρ/3 (i.e. the mass of a bubble of same radius
−2 · −2 · − − | − |
(35) butwith thedensityof thefluid)eq.42 gives a very simple
expression for theacceleration of thebubble;
Integratingover thewhole spherewe get
(m M) M R˙
Z (−∂tψ)n′dS =2πR2R˙vb+ 23πR3v˙b. (36) v˙b= (mbb+−M/2)g− 2(mb+M/2)Rvb. (43)
∂V
This is the correct version of the acceleration derived by
ThesecondintegralintheRHSofeq.34canbedirectly
Pasetto et al.(2014)intheireq.[24].Thefirstthingthatis
computedoncethevelocityfieldisevaluatedoverthesurface
apparentfrom thefirstterm in eq.43 is that,in theregime
of thesphere:
correspondingtoourphysicalapproximations,theaccelera-
v(x)=(cid:16)vbcosθ+R˙(cid:17)n′+vbsin2θeθ, at|x−rb(t)|=R(t), tcioomnpoafraedbuwbitbhletahterAesrtchisimsmedaelslerprbiyncaipflaectfoorr 1a+stMati/c(2flmuidb).
(37) While this might be surprising at first glance, its physical
explanationisquitesimple.Withintheapproximationofeq.
where we have defined the spherical coordinates r′, θ
25 the fluid is forced to be accelerated when the bubble is
(zenithalangle)andφ(azimuthalangle)measuredfromthe
accelerated.Bylookingat thestagnationpointsontopand
instantaneous center of the sphere, and e is the unitary
θ belowthebubbleitbecomesclearthatthefluidtheremoves
vectorin theazimuthal direction. From eq. 37 we get
ateverytimeatthesamevelocityasthebubble.Inorderto
2 sinθ 2 fulfillEuler’s equationfor avelocity field thatchangeswith
v(x)2 = vbcosθ+R˙ + vb , at x rb(t) =R(t). time some forces must be exerted at the boundary of the
(cid:16) (cid:17) (cid:18) 2 (cid:19) | − |
fluid (and equivalently,its reaction felt on themoving bub-
ble).Consequently,thefactor1+M/(2m )accountsforthe
b
6 Note that here the expression x means in fact at x fact that, in order to accelerate, and fulfill eq. 25, the bub-
| |→ ∞ | −
rb R. Strictly speaking the limit x is ill-defined for blemustcarrythenearbyfluidwithit.Theforceexertedon
| ≫ | | → ∞
a gravitational potential of a constant gravity field. Also, note the bubble by the surrounding medium is also responsible
that,asweareassumingthatthehydrostaticpressurechangesin
for the second term in eq. 43. In this case the term arises
muchlargerdistancesweareconsideringthatat|x−rb|≫Rthe from thefact that,asthebubbleexpands,morefluid needs
pressureP∞ depends onz sothat itcan balance the changes in tobeacceleratedtofulfilleq.25.Thistermactsinthesame
Φg(z).DuethatatthescalesoftheproblemP∞ remainsalmost
orientationasthevelocity,butitsdirectionisdeterminedby
ictonisstuasnetfu,latlsoothΦignkmtuhsetlirmemitaixn almostoncotnhsetaxnyt-.pIlnanteh,iwshceorneteΦxgt thesign of R˙. Dependingon whetherthebubbleis expand-
and P∞ are in fact strictly c|on|s→tan∞t. Then the choice of C′ =0 ingorcontracting,thistermactsinthesamedirectionasthe
correspondstochoosingΦg =−g(z−rb)−P∞(z=rb)/ρ velocityvb orintheoppositeone.Inthelattercase, itacts
MNRAS000,1–14(2015)
Bubbles, potential flows and stellar convection. 7
asadrag.Itisworthnotingthattheclaim ofPasetto et al. intheprevioussection.Theprojectedequationofmotionof
(2014) that this drag-like term reconciles the potential flow thebubblein theradial direction is
approximation with d’Alembert paradox is wrong, as this m M 1R˙ M
forceisonlypresentinthecaseofcontractingorexpanding v˙b =−mb−+ M g− 2Rm + M vb, (45)
spheres,anditisinnowayrelatedtorealdragforces,which b 2 b 2
can beof viscousor turbulentorigin. Thisis apparent from with M = 4π/3R3ρ the buoyant mass, and m the bubble
b
the fact that the force acts in the opposite direction, than mass.
thatof areal dragforce, in thecaseof contractingbubbles. To solve the bubble motion through the whole convec-
Also,itiseasytoseefromeq.21thattherelativevelocityof tive region we apply eq. 45 at a given location of a stellar
the fluid and the sphere has a tangential component at the stratification. This is the spirit of solving a problem using
surface of the sphere, contrary to what is known to happen local approach: the force balance that determines the ac-
at boundarylayers. celeration of the bubble is computed in a local approach,
Eq.43hasbeenderivedundertheassumptionthatthe and the result is used to determine the motion of the bub-
flow remains irrotational (potential) at all times. This is a blethroughtheconvectiveregion.Thismeansthatweneed
very strong physical assumption and it would be necessary to specify the value of the thermodynamic variables, T, ρ
to investigate to which extent this will be an appropriate and P, as well as their stratification given by HP, Hρ and
description of agiven realfluid.Foracompressible, viscous =dlogT/dlogP.Onlyfourofthemcanbeindependently
∇
fluid moving undera conservative body force, we havethat set, as they are related by the equation of state ρ(T,P) of
thevorticity ( v) fulfills thestellar material, which implies
∇×
D( v) dρ dP dT
∇× =(( v) )v ( v)( v) =α δ , (46)
Dt ∇× ·∇ − ∇× ∇· ρ P − T
+ ∇·τ + ∇ρ×∇P, (44) and consequently
∇×(cid:18) ρ (cid:19) ρ2
= α 1HP, (47)
where D/Dtdenotes theLagrangian derivativeandτ is the ∇ δ − δ Hρ
viscous stress tensor. In the general case, density will de-
pend both on temperature and pressure. This implies that, where α = (∂logρ/∂logP)T and δ = −(∂logρ/∂logT)P.
Inorderto solveeq.45 weneed toknowtheevolution of R
in most cases ρ P =0. Even if the flow is irrotational
∇ ×∇ 6 and M as thebubbleevolves.
atthebeginningofmotion,oneshouldexpectthatvorticity
The evolution of the buoyant mass M can be easily
( v)willbecreatedatlatertimesinarealflowbythelast
∇× obtained bytaking thetime derivativeof its definition:
termintheRHSofeq.44.Inaddition,theabsenceofadrag
forceineq.43remindsusoftheexistenceofboundarylayers M˙ R˙ ρ˙
=3 + , (48)
inrealfluidsaroundsolid bodies,whereviscosity cannotbe M R ρ
completely neglected.In boundarylayers, thethird term in
the RHS of eq. 44 will also lead to the creation of vortic- since ρ˙=dρ(r(t))/dt=−ρvb/Hρ, we have
ity. Consequently, even if the initial condition is that of an M˙ R˙ v
=3 b . (49)
irrotational flow, there is no reason to expect that the flow M R − Hρ
will remainirrotational at alltimes.Besidesthehypotheses
TheevolutionoftheradiusRofthebubblecanbeobtained
doneon theflow, thederivation of eq. 43also assumes that
from the equation of state (eq. 46) and the assumption of
the bubble remains spherical at all times. However, eq. 32
subsonicmotions. From eq. 46 it is immediate that
shows that pressure differences at the surface of the bubble
shoulddeformitassoonasitstartstomove,unlessinternal ρ˙b =αP˙ δT˙b, (50)
forces prevent it (e.g. in a solid body). Because of all these ρb P − Tb
assumptions, the use of eq. 43 to describe the movement
where we label with b the thermodynamic quantities inside
of spherical bubbles in stellar interiors might not be valid
the bubble,and we have used that P =P(r(t)).Using the
b
unless provenotherwise for each particular case.
fact that the mass of the bubble is constant, i.e. ρ˙ /ρ =
b b
Finally, up to now we have not made any assumption 3R˙/R, and using eq. 3, we finally get that the expansion
on the properties of the “bubble” element. However, in a −
of thebubbleis governed by
convection theory we want the bubble to be made of the
same material as the surrounding fluid. In the next section R˙ = δT˙b + α vb . (51)
weadoptanequationofstateforthefluidinsidethesphere R 3Tb 3 HP
and use it to describe thedynamicsof thebubble. To solve the dynamics it is still necessary to know the evo-
lution of the temperature of the bubble T . This cannot be
b
derivedwithouttakingintoaccounttheamountofheatlost
4 MOTION OF AN ISOLATED BUBBLE – (or gained) by the bubble as it moves. The energy balance
SOLUTIONS AND ASYMPTOTIC of thebubbleis given by(see Kippenhahnet al. 2012),
BEHAVIORS
dq dT δdP
=cP . (52)
4.1 General case dt dt − ρ dt
Theheat fluxF from thebubbleis given by
While it is not our aim in this paper to develop a convec-
tiontheory,wewanttoassesstheexpectedbehaviorforthe 4ac T 3
F= k T, wherek = b . (53)
motion of the bubbleunder the equation of motion derived − rad∇ rad 3 κbρb
MNRAS000,1–14(2015)
8 M. M. Miller Bertolami et al.
Estimatingthatthetemperaturegradientbetweenthebub- 4.2 Solutions for the adiabatic motion of the
ble and the surrounding fluid is dT/dR (T(r) T )/R, bubble
b
≃ −
theheat losses from thespherical bubbleare given by
Itiswellknownthatintheinnerconvectiveregionsofstars
dq 3 themovementofconvectiveelementsofreasonablesizeisal-
k (T(r) T ). (54)
dt ≃ ρbR2 rad − b mostadiabaticduetothehighdensityofthestellarmatter.
Theassumptionofadiabaticexpansiongreatlysimplifiesthe
Replacing eq.54 in eq.52 gives
treatment of eqs. 45, 49, 51 and 55. This allows for an easy
T˙ 3k T(r) v test case for the dynamics of the bubble predicted by the
b rad 1 b , (55)
Tb ≃ ρbR2cP (cid:20) Tb − (cid:21)−∇adHP method of Pasetto et al. (2014). For the sake of clarity we
will now consider the case of an ideal gas (α=δ =1) with
wP˙h=ereiPnvtbh/eHsPec,oannddtuersmedotfhtahteriagdh=th(aPnδd)/si(dcPeρwbeTbr)e.placed aInctohnestcaansteaodfiaabbatuibcbilnedmexovγin=g aΓd1ia=ba(t1ica−lly∇aind)−th1e=ste5l/la3r.
− ∇
Eqs 45, 49, 51 and 55, together with the stratification medium (k = 0) eq. 55 can be directly substituted into
rad
of the star P(r), ρ(r), T(r), HP(r) Hρ(r) and (r), allow eq.51 to give
∇
tosolvethemotionofthebubble.Thereadershouldalsobe
aware, however, that in order to use eq. 45 to describe the R˙ = vb [1 ]= vb . (56)
ad
motionofabubbleinarealflow,oneshouldfirstshowthat R 3HP −∇ 3γHP
theflowremains irrotational at alltimes. Thisisnot trivial Usingeq. 56 in eq. 49 we can derivethat
and in principlethereis noreason tostate that thegenera-
M˙ v
tion of vorticity will be small. Eqs 45, 49, 51 and 55, show = b [ ], (57)
that, even within the picture developed by Pasetto et al. M HP ∇−∇ad
(2014),itisnecessarytotakeintoaccounttheradiativeheat where we have used the fact that = 1 HP/Hρ. The
∇ −
losses from the bubble (eq. 55) before being able to solve evolutionofthebubbleintheadiabaticcaseisgivenbythe
the dynamics of the bubble. Eq. 55 shows that depending set of equations 45, 56 and 57. Note that eq. 57 describes
on the typical timescales for the expansion (τexp =HP/vb) theusualSchwarzschildcriterion.M isthemassofthefluid
andthermaldiffusion(τth=ρbcPR2/3krad)theevolutionof thatoccupiesthesamevolumeasthebubble.IfM >mbthe
Tb will be completely different. In particular, as τth R2, bubblewillriseduetobuoyancy,andifM <mb thebubble
∝
thermal diffusion always dominates the dynamics for bub- will sink due to its own weight. Let us consider a bubble
blesthataresmallenough.Intheextremecaseinwhichheat in equilibrium, i.e. M = mb, but under different values of
diffusiondominates,thebubbleexpandsinisothermalequi- ∆ = ad. When ∆ >0, a positive velocity pertur-
∇ ∇−∇ ∇
librium and there is no buoyancy. This is in stark contrast bation will lead to an increase in M, leading to an upward
withthederivationsperformedbyPasetto et al.(2014)who force (M > mb). On the other hand, a negative velocity
solve(in theirsections4and 5) thedynamicsofthebubble perturbation will lead to a decrease of M which will lead
withouttakingintoconsiderationtheroleofheatdiffusion7. to a downward force (M < mb). As expected, an unstable
It is only in theirsection 6, after havingsolved thedynam- situationresults.Similarly,∆ <0( < ad)corresponds
∇ ∇ ∇
ics of the bubble, that they consider heat losses from the toa stable situation.
bubble. We will show in the next section that solving the Substitutingeq. 56 in eq. 45, we obtain the final set of
dynamics without addressing the heat lost by the bubble equationsthat we need to solve:
can lead to extremely unphysicalresults. m M 1 M
105..I.1n0t6hesbaunldkoτfthes1o0l1a2rconηv2ecstifvoerzcoonnev,eocntievehaesleτmexepnt∼s v˙b = −mbb−+ M2 g− 6γHpmb+ M2 vb2, (58)
th
of size R ∼ ηHP (∼see Fig.×3). The motion of convective M˙ = vb ∆ . (59)
elements in those cases is very close to adiabatic down to M Hp ∇
very small sizes —i.e. η &10−3. Even in the very outer re- It is best to formulate the system using non-dimensional
gions of the sun, one finds that the expansion timescale is quantities.Wechoosetonormalizelengthswiththepressure
shorter than the thermal timescales, and the movement of scale-heightHp(=P/ρg),velocitieswiththesoundspeedcs
abubbleisclosetoadiabaticforconvectiveelementsofsize (= γP/ρ),andmasses withthebubblemassm .Inthese
b
R∼HP.Forexample,inthestandardsolarmodelofFig.3 unitps, time is measured in units of Hp/cs. The normalized
(Weiss & Schlattl2008),weseethatatr 0.999R⊙ onestill system is
fielnedmsetnhtastmτoexvpe∼alm10o3stsaadnidabτathtic∼al1ly0.6W×hη≃i2lestahnedacssounmvepcttiiovne v˙b = −γ111+−ωω − 61γ 1+ωωvb2, (60)
ofadiabaticityisgoodtostudythemotionofconvectiveel- 2 2
ementsinmostofthesolarconvectivezone,oneshouldkeep ω˙
= v ∆ . (61)
b
inmindthatitisintheregionsfarfrom adiabaticitythata ω ∇
betterconvectiontheorythanMLTisneededtopredictthe where ω = M/m . Writing m = 4π/3R3ρ , with ρ the
b b b b
correct valueof the temperaturegradient . bubbledensity,onehas
∇
ρ(r (t))
ω(t)= b . (62)
ρ (t)
7 In fact, the authors claim at the beginning of section 4.2 that b
thedynamicsofthebubbleissolvedundertheassumptionofadi- ω istheratiobetweenthebackgrounddensityandthebub-
abatic expansion. However, a careful examination of the deriva- bledensity.Wedefinethedensityperturbationofthebubble
tionsshedsthatthishypothesis isneverused. as δρ=ρb ρ,so that δρ/ρ=1/ω 1.
− −
MNRAS000,1–14(2015)
Bubbles, potential flows and stellar convection. 9
Thesystemrequirestwoinitialconditions.Thefirstini- the bubble expanded by a factor 10 100 and traveled
tialconditionistheinitialvelocity,vb(t=0);thesecondini- overroughly103Hp.Forcomparison∼,the−numberofpressure
tial condition is given byω(t=0)=ρ(r (t=0))/ρ (t=0), scaleheightintheentireSunisroughly30.Asaconclusion,
b b
the initial density perturbation of the bubble. Having nor- it is clear that the time integration has to be stopped at
malized lengthstothevalueofHP theproblemdependson some moment to make sure that the velocity of the bubble
one otherparameter, thesuperadiabaticity ∆ . remains subsonic and that the bubble did not travel out of
∇
Itisworthnotingthattheradiusofthebubbledoesnot theconvectiveregion.
enterthe adiabatic motion problem directly.However, once When δρ > 0 (dashed lines), the bubble sinks in the
asolution(v (t),ω(t))isknown,theexpansionofthebubble stratification. As a result, it contracts, and the magnitude
b
can be computed by integrating eq. 56. In normalized form ofthevelocityincreaseswithtime.Wefindthattwodifferent
it writes outcomesareobtained:thevelocitydivergeslinearlyintime
R˙ = vb. (63) ∆for ∆=∇10=−30..1, and the velocity diverges at a finite time for
R 3γ ∇
Whenthesuperadiabaticity islarge enough (∆ =0.1
∇
Wenow rewrite it as in our case, see Fig. 1), ω = ρ/ρ decreases rapidly as the
b
d r˙ bubblebecomesmoreandmoredenserthanitssurrounding.
lnR= b, (64)
dt 3γ Whenω 1, eq.60 becomes
≪
which immediately leads to v˙ = 1/γ. (67)
b
−
ln R = rb, (65) In physicalunits,this correspond to
R0 3γ
v˙ = g. (68)
b
whereR0isthebubbleinitialradius.Thechangeinthebub- −
ble radius is directly related to the distance it has traveled As nothing in the theory prevents the bubble to stop con-
from its initial position. tracting, its radius goes to zero and the bubble falls under
Eqs.60 and61 aresolved numerically.Asinitial condi- the action of gravity alone (free-fall). Its velocity diverges,
tions, we consider that the bubble is at rest, v (t=0)=0, and it becomes rapidly supersonic.
b
and we use a density perturbation to initiate the motion Whenthesuperadiabaticyissmallenough(∆ =10−3
∇
of thebubble.Weexplore positive and negativeinitial den- in our case, Fig. 2), ω = ρ/ρb does not decrease quickly
sity perturbations of different magnitudes, namely: δρ/ρ = enough,and theincrease in thevelocity magnitudenow re-
10−6, 10−3, 10−1, 0.5, 10−6, 10−3, 10−1, 0.5 —note sults in the second term in eq. 60 to be the dominant one.
−thateach−δρ/ρim−pliesa−differentδT /T sothatpressureis In thiscase, Eq 60 can bewritten as:
b b
balanced.Wealsoinvestigatedifferentvaluesofthesupera- v˙ = Cv2, (69)
diabaticity,namely ∆ =10−3,10−1.These valuescovera b − b
range going from a ne∇arly adiabatic stratification, as found whereC ispositiveandcanbeconsideredconstantintime.
in the deep stellar interior, to a value corresponding to a This gives immediately
slightsuperadiabaticity,asfoundclosetothestellarsurface 1
v (t)= , (70)
wherethemovementofthebubblecanstillbesolvedwithin b v0+Ct
theassumption of adiabatic expansion. − b
We show in Figs. 1 and 2 the solutions of the bubble where vb0 is the (absolute) value of the bubble velocity at
motion.Whenδρ<0(continuouslines),thebubbleisrising the moment where the buoyancy force becomes negligible.
and it reaches an asymptotic velocity, while ω = ρ/ρ , r , One sees from eq. 70 that the bubble velocity diverges at
b b
and log R increases continuously with time. The value of t=vb0/C.Thisisaremarkableresultthatatafirstsightmay
the asymRp0totic velocity can be derived the following way. looksurprising,yetitcanbeunderstoodinaveryeasyway
When ω 1, eq.60 becomes and shows how unphysical the predictions from the theory
≫ are.Intheextremecaseofabubblemovingadiabaticallyin
2 1
v˙ = v2. (66) an adiabatic thermal stratification (∆ =0) the buoyancy
b γ − 3γ b ∇
massand thedensitycontrast remain constant.Inthiscase
The asymptoticvelocity correspondsto v˙ =0, which leads thefirstterm intheacceleration equationremains constant
b
to vb∞ = √6. In physical units, this corresponds to √6cs. while the second one increases as the bubble increases its
This value is shown as a horizontal dashed line in the left speed. Once the second term becomes dominant the bub-
panelsofFig.1.Theasymptoticvelocityissupersonic,which ble will contract extremely fast, shrinking to a point in a
isnotconsistentwiththeunderlyingassumptionsofthethe- finite timescale. Note that, as the density contrast remains
ory. Therefore, it is clear that this asymptotic velocity can- constant to its initial value ω = ω(t = 0) this means that
not be used tocompute aconvectiveflux. at each time the bubble has sunk deep enough so that its
The timescale on which the asymptotic velocity is new densityρ (t) follows that of thebackground(ρ(r (t))).
b b
reacheddependsonlyweaklyonthemagnitudeoftheinitial In particular this implies that when R reaches R = 0 the
densityperturbation,butitdependsstronglyonthesupera- bubblehas already sunkto an infinitedepth.
diabaticity. The smaller the superadiabaticity, the longer it A particularly interesting conclusion that arises from
takes to reach the asymptotic velocity. For the largest su- the solution of the motion of the adiabatic bubble is that
peradiabaticity explored here, ∆ = 10−1, the bubble ex- there is no regime in which the acceleration of the bubble
∇
panded by a factor 10 and traveled a distance 10Hp fulfills the key eq. [41] of Pasetto et al. (2014). Not only
when it reaches the∼asymptotic velocity. For ∆ =∼ 10−3, v 2 = v˙ R but, as shown in the bottom right panels of
b b
∇ 6 −
MNRAS000,1–14(2015)
10 M. M. Miller Bertolami et al.
[t]
Figure 1.Solutionofthebubblemotionfor∆ =10−1 anddifferentinitialbubbledensityperturbations (different colorscorrespond
∇
ωto=diffMe/remntb.mBaogtntoitmudleefotfptahneepl:eretvuorlbuatitoionno).fUthpepberublebflte’psaenxepl:aenvsoioluntiaonndopfotshietiobnu.bbBloet’stovmelorcigithyt(p|vabn(etl):|)e.vUolpuptieornriogfhtthpearnaetli:oevvob2lu/t(iv˙obnRo)f.
The dashed linescorrespond to the cases whereδρ>0, forwhich vb <0, rb <0, and logR/R0 <0. Thered horizontal dashed linein
theleftupper panelshows the(conservative) limitvb=cs abovewhichthetheoryisnotvalid.Theblackhorizontal dashedlineinthe
leftpanelsistheasymptoticvelocityv∞=√6 2.45(seetext).
b ∼
Figs. 1 and 2, the ratio v 2/(v˙ R) changes over orders of derivationofthefinalequationsinPasetto et al.(2014),and
b b
magnitude during the motion of the bubble. This is a very that the key physical assumption of a rapidly expanding
strong result as this approximation is key in the derivation bubble (v /R˙ 1) is in stark contradiction with the local
b
≪
of theconvectivefluxin theirwork. and subsonic approach adopted by the authors which re-
Finally, the previous results show that the theory can- quires R/HP 1. Yet, as we haveshown in sections 3 and
≪
notbeusedtodescribethemotionofthebubbleatalltimes. 4, it is possible to solve the dynamics of the bubble consis-
The time integration has to be stopped when either one of tentlyunderthemain physicalassumption ofPasetto et al.
the quantity v , R, r reach a value where the underlying (2014),i.e.assumingadifferentiatedbubblemovinginapo-
b b
assumptions of thetheory cannot be verified anymore. tential flow. The detailed analysis of the resulting solutions
for the evolution of the bubble show a very unphysical be-
havior. This is not a surprise, as potential flows are known
to be a far-fetched idealization of real fluids. Indeed, it is
5 DISCUSSION AND CONCLUDING
known since d’Alembert that potential flows predict zero
REMARKS
drag, in strong contradiction with experience. This is the
Intheprevioussectionswehaveaddressedthetheoryofcon- famous “d’Alembert paradox” (le Rondd’Alembert 1768).
vection presented by Pasetto et al. (2014). As discussed in Potential flows are popularin text books becausethey lead
section2theirtheoryisbothalocalandatime-independent toanalytically tractableproblems.However,potentialflows
theoryofconvection,intheusualsense.Inadditionwehave are rarely achieved in real day life, and they are mainly of
shown that serious mathematical inconsistencies affect the
MNRAS000,1–14(2015)