Resonant Amplification of Turbulence by the Blast Wawes

0.46 MB·

by  A. M. Zankovich

Draftversion January18,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
RESONANT AMPLIFICATION OF TURBULENCE BY THE BLAST WAVES
A. M. Zankovich and I. G. Kovalenko
Physicotechnical Institute, VolgogradState University,Volgograd400062, Russia
Draft version January 18, 2016
ABSTRACT
6 Wediscussanideawhethersphericalblastwavescanamplifybyanon-localresonanthydrodynamic
1 mechanisminhomogeneitiesformedbyturbulenceorphasesegregationintheinterstellarmedium. We
0 considerthe problemof a blast-wave-turbulenceinteractionin the LinearInteractionApproximation.
2 Mathematically,thisisaneigenvalueproblemforfindingthestructureandamplitudeofeigenfunctions
describing the response of the shock-wave flow to forced oscillations by external perturbations in the
n
ambient interstellar medium. Linear analysis shows that the blast wave can amplify density and
a
vorticity perturbations for a wide range of length scales with amplification coefficients of up to 20,
J
withamplificationthegreater,thelargerthelength. Therealsoexistresonantharmonicsforwhichthe
5
gainbecomes formallyinfinite inthe linearapproximation. Their orbitalwavenumbersarewithin the
1
range of macro- (l 1), meso- (l 20) and microscopic (l > 200) scales. Since the resonance width
is narrow: typically∼, ∆l < 1, reso∼nance should select and amplify discrete isolated harmonics. We
]
A speculate as to a possible explanation of an observed regular filamentary structure of regular-shaped
roundsupernovaremnants suchasSNR 1572,1006or 0509-67.5. Resonantmesoscalesfound (l 18)
G ≈
aresurprisinglycloseto the observedscales(l 15)ofripples inthe shell’s surfaceofSNR0509-67.5.
. ≈
h Subject headings: hydrodynamics — ISM: supernova remnants — shock waves — turbulence
p
-
o 1. INTRODUCTION gues for the non-local nature of instability. This means
r that different fragments of the filamentary structure are
t The shapes of supernova remnants (SNRs) are gener-
s physically coupled, which provides for ordering and reg-
allyfarfromperfectsphericalgeometry. Variousphysical
a
ularity of the filamentary structure irrespective of initial
[ factors cause asphericity, among these are anisotropy of
supernova ejecta, hydrodynamic instabilities or inhomo- and, probably, environmental conditions. In the case in
1 geneities in an ambient interstellar medium. Even the point the globality of instability can be caused by the
v remnant’s size finiteness.
mostrotundremnantssuchasSNR1572and1006(Ray-
2 There are a multitude of different types of shock wave
mondet al. 2007;Winkler et al.2013)or SNR 0509-67.5
3 instabilities among which the global instabilities con-
(Warren & Hughes 2004) are endowed with small scale
8 stitute a modest set. Regarding the SNR shocks in-
structures such as ripple, filaments or knots.
3 stabilities the case that should be mentioned above all
The latter two have corrugations with a radius of cur-
0 else is the Ryu-Vishniac instability (Ryu & Vishniac
vature less than or equal to one tenth of the SNR size
1. (Raymond et al. 2007). Raymond (2003) and Raymond 1987, 1991). This instability is based on the hydrody-
0 et al. (2007) argue for the turbulent origin of the rip- namic mechanismof self-excitationof oscillationswithin
6 pling in SNR 1006 as well as in the other filamentary a spherical shock wave. However, its scope is limited by
1 SNR Cygnus Loop while Patnaude et al. (2002) speak gasesinalmostisothermalstate,theincrementsaresuffi-
: cientlysmall,andsaturationoccursatweaklynon-linear
v for multiphaseness.
level (the density enhancements have a factor of . 2)
i Even accepting the hypothesis of turbulent or multi-
X (Mac Low & Norman 1993).
phaseoriginwehavetoadmitatthesametimeanamaz-
The spherical shock in the Ryu-Vishniac representa-
r ingregularityofpatternimprintingfineripplingthatcan
a tion can be considered a coherent oscillator generating
be seen on the recently published composite optical-X-
ray image of SNR 0509-67.5 from Hubble and Chandra self-sustained oscillations. The spherical shock again is
a sphericalresonantcavity that can be excited by exter-
releasedbyNASA(2012). Onecanrecognize 15knots
∼ nal disturbances. Indeed, the environment for the inter-
observed in longitude: this regular speckled structure
stellar shock is not in the least a perfect uniform back-
envelops the whole remnant’s surface. This regular pat-
groundbutusuallyastronglyinhomogeneous,turbulized
ternsuggeststhatsomelengthscalesofperturbationsare
or phase-separated medium.
morepreferredthanothersandthisishardtounderstand
Inthe presentpaperweanalyzethe efficiencyofpossi-
in a turbulence scenario that is scaleless by nature.
ble resonantamplificationof disturbances by the spheri-
Quite the reverse, it would appear reasonable that
cal shock wave. The basic idea is that a supernova rem-
there exists a physical mechanism amplifying few inter-
nant can filter out some resonant length scales within
mediate scales. A plausible explanation would be that
chaotic and globally scaleless noise in the ambient tur-
we observe the development of some instability on the
bulent or phase-separated interstellar matter.
shock surface of the remnant. If the idea of instability
The possibility of turbulence amplification by the
is correct, the fact that the excited scales are not van-
shock waves has been a debatable issue for a long
ishingly small but comparable to the remnant’s size ar-
time. Such a possibility has been proved theoretically
[email protected] (Wouchuk et al. 2009; Donzis 2012),numerically (Lee et
2 Zankovichand Kovalenko
al. 1997; Balsara et al. 2001; Jamme et al. 2002) and ∂(ρv) +div(ρvv)+ p=0, (2)
experimentally (Andreopoulos et al. 2000). ∂t ∇
The analyticalstudies focus mainlyonthe localmech-
∂ ρv2 p ρvv2 γpv
anisms of turbulence amplification. However, when the + +div + =0, (3)
finiteness of the flow becomes significant,non-localreso- ∂t 2 (γ 1) 2 (γ 1)
(cid:18) − (cid:19) (cid:18) − (cid:19)
nant effects can appear.
which should be complemented by the outer boundary
In our study we rely on the Linear Interaction Ap-
conditions at the strong shock front which in our case
proximation which is mainly used since the pioneering
read as
work by Ribner (1954) to analyze the shock-turbulence
ρ V=ρ (v V), (4)
interaction. Such an approximation seems to be reason- − 0 s s−
ableifthe turbulence issignificantlysubsonicandalow- ρ V2 =ρ (v V)2+p , (5)
amplitude one, which of course is the case under consid- 0 s s− s
eration. 1 1 γ p
Toacertainextent,ouranalysisresemblesanapproach V2 = (vs V)2+ s . (6)
2 2 − γ 1ρ
of (Sari et al 2012); however, it possesses greater versa- − s
tility. Hereρisthedensity,vthevelocity,pthepressure,andV
The outline of the paper is as follows: in Section 2 we isashockfrontvelocity. Hereafterwedenotebythesub-
describe the physical model of the shock wave produced scripts “0”, “1” the states outside and inside the shock-
bythepointexplosion,discussitssymmetrical(scaleless) wave flow, respectively, and “s” denotes the states just
properties,definesimilarityvariablesbothfortheunper- behind the shock front.
turbedflowandperturbations,andposeamathematical Furtherweconsiderbothpreshockandpostshockflows
problem for the postshock perturbations with particu- as a superposition of the unperturbed (mean) flow and
larfocus onderivationofthe inner boundaryconditions. small perturbations
Mathematically, the computational task is reduced to
ρ=ρ +δρ , v=v +δv , p=p +δp . (7)
an eigenvalue problem for finding the amplitudes of the 0,1 0,1 0,1 0,1 0,1 0,1
postshockperturbationsiftheamplitudes andfrequency Theparametersofthe meanambientmediumarecon-
ofouterperturbationsaregiven. InSection3wepresent sidered as a motionless, zero-pressure state with spheri-
the results of numerical integration. We show that both cally symmetric power-law density distribution
for the vortex modes and for the entropy modes there
existconditionsforinfinite(inthelinearapproximation) ρ0 =A/rω, v0 =0, p0 =0, (8)
resonant amplification. In Section 4 we discuss the do-
where r is the sphericalradius, A and ω arecertain con-
main of applicability of solutions found and present a
stants.
brief summary.
We suppose that at the initial moment t = 0 an in-
stant release of energy E takes place in the origin that
0
2. THEMODEL generates a flow with a strong spherical shock wave.
2.1. Basic Equations and Unperturbed Flow The mean values inside the shock flow obey the self-
similarSedovsolution(Sedov1959)inwhichthegovern-
AllsupernovaremnantsmentionedinIntroductionare ing parameter, the similarity index δ is equal to
young or middle-aged objects with sizes 10-20 pc for
whichapproximationofadiabaticexpansio∼nholds. Thus 2
δ = , (9)
we base our consideration on an adiabatic shock wave 5 ω
model. −
characterizesthe rateofthe unperturbedshockfrontex-
The magnetic field becomes dynamically important if
pansion,
magnetic pressure becomes comparable with dynamic
and/orthermodynamic pressure. Simple estimates show dR R
that magnetic pressure becomes equal to the density of Rs(t) tδ, V = s =δ s . (10)
∼ dt t
released supernova energy 1051 erg smeared out all over
the spherical volume with a radius of 10 pc if the mag- Further we use variables for the mean flow
netic field attains the value 300µG. There are many ρ v p
∼ ρ˜= 1 ρ˜(ξ), v˜= r1 v˜(ξ), p˜= 1 p˜(ξ), (11)
magnetized remnants, and SN1006 with 170µG(Mar- ρ ≡ v ≡ p ≡
cowith&Casse2010)amongthem,forwh∼ichB-fieldcan- s rs s
not be neglected. On the other hand, Warren & Hughes normalized by the postshock values
(2004)estimatemagneticfieldvalueinSNR0509-67.5as
γ+1 2V 2ρ V2
60 µG; therefore magnetic pressure is insignificant there ρ =ρ , v = , p = 0 , (12)
at the adiabatic stage at least. In the present paper we s 0γ 1 rs γ+1 s γ+1
−
limit ourselves to an examination of the pure hydrody-
and the self-similar variable as the radial coordinate
namical model neglecting the effects of magnetic pres-
r
sure. ξ = . (13)
The shock wave dynamics is described by the system Rs(t)
of gasdynamic equations for inviscid perfect gas with an
In dimensionless variables (9)-(13) the system (1)–(3)
adiabatic index γ
for the mean flow assumes the form
∂ρ 2 dρ˜ 2 dv˜ 4 v˜
+div(ρv)=0, (1) v˜ ξ + + ω =0, (14)
∂t γ+1 − ρ˜dξ γ+1dξ γ+1ξ −
(cid:18) (cid:19)
Resonant Amplification of Turbulence by Blast Waves 3
2 dv˜ γ 1 dp˜ 3 ω This new coordinate system allows splitting the nabla
v˜ ξ + − − v˜=0, (15)
γ+1 − dξ γ+1ρ˜dξ − 2 operatorin expansions similar to (20) to radial, , and
(cid:18) (cid:19) ∇ξ
tangential, ,partsandtocomplementitbythebinor-
τ
2 dρ˜ 2 dp˜ mal counter∇part :
γ v˜ ξ v˜ ξ +(3 γω)=0, ∇b
γ+1 − ρdξ− γ+1 − p˜dξ −
(cid:18) (cid:19) (cid:18) (cid:19) (16) =e ∂ , = = eθ ∂ + eϕ ∂ ,
Onehas twoqualitativelydifferentkinds ofthe unper- ∇ξ ξ∂ξ ∇τ ∇−∇ξ ξ ∂θ ξsin(θ)∂ϕ
turbedflow,theshell-likeonewithahollowcavityinside e ∂ e ∂
(ω > 7−γ) or the solid flow which extends to the cen- b =eξ = ϕ θ .
γ+1 ∇ ×∇ ξ ∂θ − ξsin(θ)∂ϕ
ter of symmetry (ω < 7−γ) (Sedov 1959). In physical (19)
γ+1
applications the most important are the cases of either
uniform background ω = 0 or the power-law decreasing The natural system optimizes the analysis inasmuch
density with the slope ω > 0. In what follows we as- as the binormal component of perturbed velocity de-
sume 0 ω < 3. The latter inequality guarantees mass generates and thus perturbations are reduced to two-
≤
integrability in the origin. dimensional ones.
Ultimately, the symmetry and similarity of the flow
2.2. Perturbations allow us to look for a solution for perturbations as an
expansion in the solid spherical harmonics
In describing perturbations we also use the nondimen-
sionalized form:
∞ l
f = δρ0,1, f = δvr0,1, f = δvτ0,1, fρ(ξ,t)= fρlm(ξ)Ylm(θ,ϕ)t−isl,
ρ0,1 ρs vξ0,1 vrs vτ0,1 vrs (17) Xl=0mX=−l
fvb0,1 = δvvbr0s,1, fp0,1 = δpp0s,1. fv(ξ,t)=Xl∞=0mX=l−l(cid:16)fvξlm(ξ)Ylm(θ,ϕ)eξ
Heresubscriptsτ andbstandfortangentialandbinormal
components of velocity to be explained further and the +fvτlm(ξ)ξ∇τYlm(θ,ϕ)+fvblm(ξ)ξ∇bYlm(θ,ϕ) t−isl,
subscript “1” for the postshock perturbations is further ∞ l (cid:17)
omitted. f (ξ,t)= f (ξ)Y (θ,ϕ)t−isl.
The symmetry of the flow permits looking for a so- p plm lm
lution for perturbations as an expansion in the vector Xl=0mX=−l
(20)
spherical harmonics. One can find different definitions
of the vector spherical harmonics in the literature (Bar-
Hereξ=(ξ,θ,ϕ)hasameaningoftheradiusvectorina
rera et al 1985; Winch et al. 2005); we follow that of
comoving spherical coordinate system, s is a frequency
(Kovalenko & Eremin 1998). l
inthe‘comoving’logarithmictimescalet˜=log(t/t )and
Heuristic arguments for this expansion are as follows. 0
thefrequencydoesnotdependonmsincethedifferential
The sphericity of the unperturbed flow prompts seek-
equations for perturbations do not depend on the m-
ingtheexpansionofscalarfunctionsinaformoffactored
wavenumber.
out terms f(ξ)Y (θ,ϕ) known as solid spherical har-
lm
∼ The radial dependencies of perturbations f (ξ) are
monics. Spherical functions Y (θ,ϕ) compose a com- i
lm
found by solving the system of linearized hydrodynamic
plete orthogonal basis in the space of functions defined
on a sphere S2. equationswhichcanbeobtainedfromthesystem(1)-(3)
The vector functions are expanded as follows. The with the help of (11)-(13),(17),(20). Orthogonality of
geometry of the problem defines three preferred direc- harmonics allows uncoupling and factorization of equa-
tions. Twoofthemaredeterminedthroughthe gradient tions (Ryu & Vishniac 1991):
of solid spherical harmonics (ξnY (θ,ϕ)): the first,
lm
∇
radial one, is given by the gradient of the radial part, γ+1 df df
e = ξ, the second direction is determined by the gra- v˜ ξ ρ +ρ˜ vξ
ξ − 2 dξ dξ
dient∇of the spherical harmonic, Y (θ,ϕ). By defini- (cid:18) (cid:19)
lm
tion the latter one is tangential to∇the concentric sphere dv˜ v˜ γ+1 1
+ +2 ω (γ+1)(5 ω)is f
of radius ξ and thus orthogonal to e . The third direc- dξ ξ − 2 − 4 − ρ
ξ (cid:18) (cid:19)
tionis thendetermined unambiguouslyas anorthogonal
dρ˜ ρ˜ ρ˜
complement. + +2 f l(l+1) f =0, (21)
dξ ξ vξ − ξ vτ
The right-hand triple of unit vectors (cid:18) (cid:19)
e = (ξ),
ξ
∇
Y (θ,ϕ) γ+1 df γ 1df γ 1dp˜
e = ∇ lm , l2+m2 =0, v˜ ξ ρ˜ vξ + − p − f
τ ρ
Y (θ,ϕ) 6 (18) − 2 dξ 2 dξ − 2ρ˜ dξ
|∇ lm | (cid:18) (cid:19)
e Y (θ,ϕ) dv˜ 1 1
e = ξ×∇ lm . + (γ+1)(3 ω) (γ+1)(5 ω)is ρ˜f
b |eξ×∇Ylm(θ,ϕ)| (cid:18)dξ − 4 − − 4 − (cid:19) vξ
=0, (22)
thus sets the ‘natural’ orthogonalsystemof coordinates.
4 Zankovichand Kovalenko
v˜ γ+1ξ ρ˜dfvτ For simplicity, we further limit ourselves to the prin-
− 2 dξ cipal mode n = 0, the subscript n will, therefore, be
(cid:18) (cid:19)
omitted. It follows therefrom that the absolute exter-
dv˜ 1 1
+ (γ+1)(3 ω) (γ+1)(5 ω)is ρ˜f nal perturbations δρ , δv , δp under consideration are
dξ − 4 − − 4 − vτ 0 0 0
(cid:18) (cid:19) spatially uniform.
γ 1 Assumingtheturbulentmotionintheambientmedium
+ − f =0, (23)
2ξ p to be a slow and subsonic one, and the outside pres-
sureperturbationsnegligiblecomparedwiththepressure
behind the strong shock wave, we can neglect the ex-
v˜ γ+1ξ dfvb ternal acoustic perturbations (fp0 → 0) and consider
− 2 dξ just two different modes, either (i) the vortical mode
(cid:18) (cid:19) (f = 0, f = f = 0), or (ii) entropy perturbations
+ ddvξ˜ − 14(γ+1)(3−ω)− 41(γ+1)(5−ω)is fvb t(ficvρ00p6e6=rtu0r,baρft0vi0on=s p(0ffp0 == 00)).caInnnoatnybecacosen,sitdheeredacoseulsf--
(cid:18) (cid:19) p0
6
=0, (24) consistently as far as they violate the similarity condi-
tion.
As we know, the vortex and entropy modes are frozen
γ+1 dfρ γ+1 ρ˜dfp in the environment which we regard as motionless, thus
γ v˜ ξ + v˜ ξ
− − 2 dξ − 2 p˜ dξ they have zero frequency in the rigid reference frame.
(cid:18) (cid:19) (cid:18) (cid:19)
At the same time, relative perturbations defined in the
1dp˜ γ dρ˜
+ ρ˜f comovingframeandnormalizedaccordingto(17)should,
p˜dξ − ρ˜dξ vξ of course, be time-dependent.
(cid:18) (cid:19)
γ+1 1dρ˜ 1 Stationary preshock perturbations must obey the lin-
+γ v˜ ξ + (γ+1)(5 ω)is f
ρ earized continuity equation
− 2 ρ˜dξ 4 −
(cid:18)(cid:18) (cid:19) (cid:19)
γ+1 1dp˜ 1 ρ˜ div(ρ δv )=0. (27)
v˜ ξ + (γ+1)(5 ω)is f 0 0
p
− − 2 p˜dξ 4 − p˜
(cid:18)(cid:18) (cid:19) (cid:19) Taking into account expansion in spherical harmonics,
=0. (25) withtheaidof (8)wegetfrom(27)therelationbetween
the radial and tangential velocity components
Hereinafterthesubscriptslandmareomitted. Thetypo
inthe bracketedexpressionatthefirsttermin(25)from
l(l+1)
(Ryu & Vishniac 1991) is corrected as well. δv = δv . (28)
ξ0 2 ω τ0
Itisworthnoting oncemorethatthe system(21)-(25) −
is m-wavenumber degenerated and that the variable fvb Thisisauniqueconstraintfortheoutsideperturbations:
uncouples (Eq. (24)). theotherlinearizedhydrodynamicsequationsforthemo-
mentum and energy conservationare satisfied automati-
2.3. External Perturbations cally.
The system (21)-(25) describes perturbations both in- Thebinormalcomponentofvelocitydoesnotenterthe
side and outside the shock wave. First consider the ex- equation (27) and can be nullified so long as it does not
ternalto the shockperturbations codedby the subscript couple with other terms due to neither boundary condi-
“0”. tions on the shock front nor hydrodynamic equations.
The radial function f(ξ) in solid harmonics is usually In total, one of three perturbations δvr0lm, δvτ0lm,
chosen as an expansion in eigenfunctions of some oper- δρ0lm can be considered free whereas the other two
ator relevant to the problem (Landau & Lifshitz 1991). should be dependent ones. For the entropy mode we
For example, for the Laplace equation assume
f =0, f =f =0, (29)
∆φ=0 ρ0 6 vξ0 vτ0
for the vortical mode we have
the radial parts are just power functions; the expression
∞ l f =0, f = l(l+1)f , f =0. (30)
φ= φ ξnY (θ,ϕ) ρ0 vξ0 2 ω vτ0 vτ0 6
lmn lm −
nX,l=0mX=−l By virtue of stationarity of absolute perturbation,
yieldsageneralsolutionoftheLaplaceequationinaball. δρ0 t0, and self-similarity, ρs t−δω, (see equations
∝ ∝
This power-law expansion of the radial function is (8), (10)) for the relative perturbation we get
nothingbuttheTaylorseries. Consideringtheperturba-
δρ γ 1δρ
tions as a regular scalar or vector field bounded within f = 0 = − 0 tδω. (31)
the finite domain we use the power-law solid spherical ρ0 ρs γ+1 ρ0 ∝
harmonics
With regard to proportionality f t−is according to
∞ ρ0 ∝
(20) and using the definition (9), we finally find the fre-
f (ξ)= f ξn, i=(ρ,v ,v ,v ,p) (26)
i0 i0n ξ τ b quency for the entropy mode
n=0
X
2ω
as a basis for the external perturbations expansion. s(e) = i. (32)
5 ω
−
Resonant Amplification of Turbulence by Blast Waves 5
For the vortex mode with the aid of relations δv t0, In fact the condition at the inner boundary, as well as
0
v tδ−1 we similarly get ∝ theouterboundary,isactuallyano-sourceconditionand
s
∝
canbe explicitlyderivedfromEquations(1)-(3)express-
3 ω
s(v) = − i. (33) ing conservation laws.
5 ω Letusintegratethelinearizeddivergenttermsfromthe
−
Equations(1)-(3)overthesmallsphereS ofvanishingly
Thefrequencies(32),(33)canequallywellbefoundupon ǫ
small radius ǫ and the volume V centered at the origin.
substituting (29), (30) into (21)-(25). ǫ
For the linearized mass flux we get
2.4. Perturbations Inside the Shock Wave
δI = div[δ(ρv)]dV = (δρv +ρ δv)dS
The radial dependencies f (ξ) of the solution for the mass 0 0
i
perturbationsinsidethe shockwavearenotpresumedto VZǫ SIǫ (40)
be power series expansion; rather, they are to be found
by solvingthe system(21)-(25)with the inner andouter =(δρ(ǫ)v0(ǫ)+ρ0(ǫ)δvξ(ǫ))ǫ2 YlmdΩ,
boundary conditions. I
4π
2.5. Boundary Conditions for Perturbations where Ω stands here for the solid angle.
In a similar manner for the fluxes of momentum and
2.5.1. Outer Boundary Conditions
energy we have
Linearized boundary conditions for the amplitudes of
perturbationscanbededucedfromthegeneralboundary δI = div δ ρvv+pIˆ dV
mom
conditions (4)-(6) with regard to the fact of the shock
surface warping: VZǫ h (cid:16) (cid:17)i
dρ˜ γ+1 = δρ(ǫ)v2(ǫ)+2ρ (ǫ)v (ǫ)δv (ǫ)+δp(ǫ) ǫ2 Y e dΩ
f = ω+ η+ f , (34) 0 0 0 ξ lm ξ
ρs −(cid:18) dξ(cid:12)s(cid:19) γ−1 ρ0 (cid:0) (cid:1) 4Zπ
(cid:12)
(cid:12) dp˜ +ρ (ǫ)v (ǫ)δv (ǫ)ǫ3 Y dΩ,
f = 2 ω (5(cid:12) ω)is η 0 0 τ ∇τ lm
ps (cid:18) − − − − dξ(cid:12)s(cid:19) (35) 4Zπ
γ+1 4 (cid:12) (41)
+ f f ,(cid:12)
γ−1 ρ0 − γ+1 vξ0 (cid:12) δI = div δ ρvv2 + γpv dV
(5 ω) dv˜ γ 1 ener 2 γ 1
f = 1 − is η+ − f , (36) Z (cid:20) (cid:18) − (cid:19)(cid:21)
vξs − 2 − dξ γ+1 vξ0 Vǫ
(cid:18) (cid:12)s(cid:19) 1
fvτs =−η+(cid:12)(cid:12)(cid:12)fvτ0, (37) =2 δρ(ǫ)v03(ǫ)+3ρ0(ǫ)v02(ǫ)δvξ(ǫ) ǫ2Z YlmdΩ (42)
(cid:2) (cid:3) 4π
fvbs =fvb0 =0. (38) + γ [δp(ǫ)v (ǫ)+δv (ǫ)p (ǫ)]ǫ2 Y dΩ.
0 ξ 0 lm
γ 1
Corrugation of the shock front is described here by the
− Z
4π
small dimensionless parameter
The vector integral in Equation (41) Y e dΩ is
∆R lm ξ
η(θ,ϕ,t) s = η Y (θ,ϕ)t−isl 1, (39) 4π
≡ Rs lm lm ≪ nonzero only for the dipole l = 1, the sRcalar integral
Xl,m Y dΩ in Equation (40) and (42) becomes zero in all
lm
where ∆R (θ,ϕ,t) is a small shift of the front from its 4π
s iRnstances except for the monopole l = 0. Although the
mean value R (t).
s fluxes integrated over full solid angle can vanish identi-
2.5.2. Inner Boundary Condition at the Center of Symmetry cally due to the specific nature of the spherical harmon-
ics, the sectorial flux density may not.
The relationship between characteristics requires only As an inner boundary condition we demand that the
one inner boundary condition. flux must vanish in each direction, that is,
Itisnotablethatderivationofthecorrectinnerbound-
arycondition has a long history. Ryu & Vishniac (1987) lim f (ξ)v˜(ξ)+ρ˜(ξ)f (ξ) ξ2 =0, (43)
derived the equation δp(0) = 0 which was criticized by ξ→0 ρ vξ
Kushnir et al. (2005). Indeed, the condition of null per- (cid:2) (cid:3)
lim f (ξ)v˜2(ξ)+2ρ˜(ξ)v˜(ξ)f (ξ)+f (ξ) ξ2 =0, (44)
turbed pressure seems to go beyond the bounds of com- ξ→0 ρ vξ p
monsense. Ifweuniformlycompressanelementoffluid,
(cid:2) (cid:3)
pressure will definitely grow inside the fluid. Kushnir et limρ˜(ξ)v˜(ξ)f (ξ)ξ2 =0, (45)
al.(2005)proposedanotherconditionwhichtheyformu- ξ→0 vτ
lated in anasymptotic form suggestinga lack ofsimilar-
ity at the origin due to specific initial conditions (non- lim 1f (ξ)v˜3(ξ)+ 3ρ˜(ξ)v˜2(ξ)f (ξ)
point and probably non-instant energy release). This ξ→0 2 ρ 2 vξ
conditionbeingadaptedfornumericalcalculation,noan- (cid:20) (46)
γ
alytical expression was presented. + f (ξ)v˜(ξ)+f (ξ)p˜(ξ) ξ2 =0.
γ 1 p vξ
− (cid:21)
(cid:0) (cid:1)
6 Zankovichand Kovalenko
The very unique inner boundary condition is the one cumbersome; its existence can be checked numerically.
majorizing term in Equations (43)-(46) that guarantees However we manage without its explicit expression by
the fulfillment of all four conditions. To find this ma- specifying conditions directly to amplitudes.
jorizing term we have to determine the relation among Substituting Equations (48), (49) to the boundary
perturbed variables. fluxes (43)-(46), we find the asymptotics ξ3+α for the
First, we note that the unperturbed variables asymp- mass flux and all terms in (43) are of the same order.
totically behave at the origin (ξ 0) as (Sedov 1959) The asymptotics for the momentum fluxes is ξ4+α and
→
ρ0 =a0ξa−2, v0 =b0ξ, p0 =c0+c1ξa, nalalltlyerfmorstihne(4en4)e-r(g4y5)fluaxretohfetahseysmampteotoircdseirsaξg5−aian+.αAwnhdicfih-
1+γ(2 ω) is determined by the dominating term f ξ2. The ma-
a= − , vξ
γ 1 jorizingtermamongthese is thatwhosepowerexponent
− is the minimum one. If a < 2, the mass flux dominates,
2(3 ω) (1+γ)ω 6
κ = − , κ = − , if a > 2 the energy flux does. In the latter case the
1 2
(ω 5)(γ 1) 6 3γ ω majorizingterm is f ξ2, in the former case vanishing of
− − − − vξ
γ2(ω2 4ω+13)+γ(ω+1)(ω 7) 2ω+12 f ξ2 implies vanishing of all other perturbation terms.
κ = − − − , vξ
3 (ω 5)(3γ+ω 6)(3γ 1) Thus, the inner boundary no-source condition yields
− − − an equation
a0 =(cid:18)12+γγ(cid:19)κ1(cid:18)1+γ γ(cid:19)κ2 ωω−−271γ+−γ+γγ1!3γ−−ω1κ3, fvξξ2(cid:12)(cid:12)ξ=0 =0. (52)
c0 =(cid:18)12+γγ(cid:19)5−6ω (cid:18)1+γ γ(cid:19)κ2+1 ωω−−271γ+−γ+γγ1!3κ3, d2e.Ir5ni.v3ne.edrInbanoseurtnhBdeoaurcynodncadorinytdioCitnoionf(cid:12)ndoirtfiooarntcfhooenrtstahhceetlFl-illnoiktweerwflfaiotchweHcinaolnltobhwee
b0 = γ2+γ1, c1 =a0b20γγ((52−ωω))−+21. pdiotiinotnsξimnu>st 0besefpualfirlaletidngsimgausltaanndeouvaslcyuufomr.thTewcoonctoanct-
− discontinuity: (i) there is no flux through the shifted
(47)
boundary and (ii) the pressure at the shifted boundary
Here a > 1 for 0 ω < (7 γ)/(γ + 1) and γ > 1. isidenticallyzero. Equationsforthesecanbepractically
This particularly m≤eans that t−he term dp˜/p˜dξ is asymp- derived from the general matching conditions (4)-(6) by
totically negligible comparedto other terms in Equation nullingl.h.s. andbysubstitutingthedisplacementofthe
(16) as well as in Equation(25) but the other term with inner boundary ηin for η and the subscript “in” for “s”.
pressure ρ˜/p˜and the pressure gradient itself matter. We obtain two conditions for the velocity and for the
Thecoe∼fficientsofthesystem(21)-(23),(25)asymptot- pressure (Ryu & Vishniac 1991)
icallybehaveaspowerfunctionsattheorigin. Therefor,e
welookforsolutionsforperturbationsaspowerlawsinξ. γ+1 is dv˜ξ
fFrroommEEqquuaattiioonn((2212))wweefignedt favlξso∼fξp3−∼afξρa,−f1vfτvξ∼∼ξ3−ξa2ffρρ.; fvξ =− 2 + 4(γ+1)(5−ω)+ dξ (cid:12)(cid:12)(cid:12)ξin!ηin(5,3)
Equations (23) and (25) confirm relations found. dp˜ (cid:12)
Suppose f = η . (54)
p in
fρ =Fρ0ξα (48) −dξ(cid:12)ξin
(cid:12)
withsomeconstantamplitudefρ0,then,similarly,weget Accurate within a misprint th(cid:12)(cid:12)ese formulae coincide with
thosededucedbyRyu&Vishniac(1991). Eliminationof
f =F ξ3−a+α, f =F ξ3−a+α, f =F ξ2+α. η setsseeminglydesiredrelationbetweentwounknown
vξ vξ0 vτ vτ0 p p0 in
(49) quantities.
The solution for perturbations then can be assembled in A more thorough analysis shows however that while
a vector form as (dv˜ξ/dξ)|ξin inEquation(53)remainsfiniteandnonzero,
thederivative(dp˜/dξ) inEquation(54)caneitherdi-
f =F (ξ)ξα, F (ξ)=(F ,F ξ3−a,F ξ3−a,F ξ2)T. |ξin
0 0 ρ0 vξ0 vτ0 p0 verge or vanish. Indeed, we know from Sedov (1959) an
(50) asymptotical behavior of unperturbed pressure near the
Substituting Equations (48), (49) into Equations (21)- inner boundary
(23), (25)we getthe systemofhomogeneouslinearalge-
braic equations for the amplitudes (Fρ0,Fvξ0,Fvτ0,Fp0), p˜(ξ)≈p˜0·(ξ−ξin)µ+1, p˜0 =const, (55)
from which we find four roots for α as the eigenvalues.
where µ determined as
The asymptotic solution is ultimately a superposition of
four branches with some amplitudes Ci ω(1+γ) 6 7 γ
µ= − , − ω <3, (56)
4 6 3γ ω γ+1 ≤
f = C F (ξ)ξαi. (51) − −
i 0i reverses sign in the tolerance interval of ω. Thus, Equa-
Xi=1 tions (53)-(54) in reality are a singular system.
One of the root, say α , describes a divergent solution The correctcondition canbe derivedby exceeding the
1
for a point sourcewhile three other arethe regularones. limits oflinearapproximationandbyusing the property
The analytical expression for this exponent α is quite (ii) of the contact discontinuity. Since from now on, we
1
Resonant Amplification of Turbulence by Blast Waves 7
1 2
0.5 1.5
0
−00.5l= s=0 0.51 γω==42/.37 l≈18 l≈80 l≈231 l≈599
m(s) m(s) 0 im(s)≈ 0.1304
I −1 I
−0.5
−1.5
−1
−2
−1.5
−2.5 −2
10−1 100 101 102 10−1 100 101 102 103
l
l
Figure 1. The image part of dispersion curves (frequency s vs Figure 2. The same as in Fig. 1 but for the nonuniform back-
orbital wavenumber l) for the blast wave free oscillations in the
ground(ω=2.7,γ=4/3).
case of the uniform background (ω = 0, γ = 1.1. Positive Im(s)
correspondtounstableoscillations. Dashedlinesindicatelocation
oftheresonancepoint. Zeroizingtheamplitudesoftheexternalperturbations,
one finds the structure of the perturbed flow inside the
temporally use non-linear dependencies, let us tempo-
shockwavebysolvingintheintervalξ ξ 1,ξ 0
rallyconsiderallperturbationsasrealvariables. For the in ≤ ≤ in ≥
′ the system of linear ODEs (21)-(23),(25) for unknown
perturbed pressure p˜ at the perturbed inner boundary
1 variables f (ξ), f (ξ), f (ξ), f (ξ), η with the outer
in the first order we have vρ vξ vτ p
(35)-(37) and inner (52), (59) or (60) boundary condi-
0=p˜′(ξin+ηin)=p˜0·(ηin)µ+1+fp, ηin >0; (57) tions. The frequency s of free oscillations is then found
0=f , η <0. as an eigenvalue.
p in
The typical dispersion dependencies of s on the wave
Taking into account Equations (53), (54), (55), (57), we number l for the initial few lower order modes are pre-
finally get a non-linear relation: sented in Figures. 1 (solid flow, ω = 0, γ = 1.1) and 2
(shell flow, ω = 2.7, γ = 4/3). Each mode meets a par-
f fµ+1, η >0. (58)
p ∝ vξ in ticularnumberofnodesofoscillationsalongradius. The
Inthe caseµ>0,consideringthe vanishingamplitude upper mode in Figures. 1 and 2 is the principal mode
limit, f 0, we find f f or asymptotically (no nodes), the other ones, from top downward, higher
vξ → | p|≪| vξ| order, so-called reflection modes have 1, 2 and so on in-
definitelynodes. Theflowwithγ closeto1approximates
f =0. (59)
p ξin the radiative flow to some degree, while γ close to 5/3
(cid:12) pertains to pure adiabatic flow.
For µ<0 the situation is(cid:12)reversed ∗
(cid:12) Suppose the external perturbation has frequency s
∗
andthewavenumberl . Itwillbetheresonantoneifthe
fvξ =0. (60) point (s∗,l∗) is disposed exactly on the dispersion curve
ξin
(cid:12) s(l) (represented by dashed lines in Figures. 1 and 2) or
In the particular case µ =(cid:12) 0 we come back to the initial ∗
(cid:12) initsclosevicinity. Wewouldliketoremindthats and
Equations (53)-(54).
s(l) take on the complex values in the general case and
∗ ∗
hence(s ,l )ands(l)mustcoincidenotonlyintheimag-
3. AMPLIFICATIONOFDISTURBANCESBYTHESHOCK
inarypartsasisdisplayedinFigures.1-2butalsoinreal
WAVE
parts. Since we have adopted the uniform distribution
In order to know whether the shock wave amplifies ∗
of external perturbations, their frequencies s can only
perturbationsornot,wehavetofindthestructureofthe
beimaginaryonesaccordingtoEquations(32),(33)and
perturbed postshock flow. We expect amplification first
non-negative if ω 0. Then the equality of frequencies
of all in the case of resonance between the free (i.e. self- ≥
is possible in the increments domain where Ims(l) 0.
excited) oscillations of the flow and oscillations excited ≥
Numerical linear analysis (Ryu & Vishniac 1987, 1991)
by the external perturbations if resonance only exists.
shows that the unstable regimes exist at γ <1.2 for the
solidflow (ω =0) and for any permitted γ in the case of
3.1. Self-excited Oscillations and Resonance Conditions
the shell flow.
Theeigenmodesoftheshockflowwerestudiedindetail The condition for resonance
byRyu&Vishniac(1987,1988);Vishniac&Ryu(1989),
∗ ∗ ∗
Kushniretal.(2005)forthefilledflowandbyGoodman Res(l )=0, Ims(l )=s 0 (61)
≥
(1990)andRyu&Vishniac(1991)forthe shell-likeflow.
is fulfilled for the entropy mode in the case of the solid
To find the resonance conditions we calculated first the
flow only in the origin at the point
dispersionrelationsforfreeoscillationsguidedbytheap-
∗ ∗
proach of the aforementioned authors. s =0, l =0; ω =0, (62)
8 Zankovichand Kovalenko
whereas the unstable section 101 . l . 102 (Figure 1)
corresponds to nonzero Res(l). 30
In the shell case the resonance frequency for the en- 2 fp
tropy mode cuts the sequence of points with high wave
1
numbers 20 fv fvo
s∗ =i·2.347..., l∗ ≈230, 700,...; ω =2.7; (63) fp 0 f fvrfvfroo fpo
they are located beyond the object area in Figure 2. 10
0.99 1 1.01
The resonance condition for the vortex mode
s∗ =0.6i; ω =0, (64) fv fpo f
f 0
is never fulfilled for the solid flow by the same token fvr fvro
f
Res(l) = 0 but is fulfilled at growing unstable branches
f(oFrigtuhrees6 h2e)llflowandspecifiesanothersequenceofpoints 0.20 fvτ fvτο
∗ ∗ −0.2
s =i 0.1304..., l 18,...; ω =2.7. (65)
· ≈ −0.4
3.2. Forced Oscillations. Amplification of Disturbances −0.6
behind the Shock Front −0.8
0.6 0.8 1
In order to find the structure of the postshock flow 0.5 0.6 0.7 0.8 0.9 1 1.1
perturbed by external disturbances, we have to find
the eigenfunctions considering the arising mathematical
problem as an eigenvalue problem. Figure 3. The real parts of the eigenfunctions for the density,
radial velocity and pressure perturbations (solid lines), the tan-
Given the amplitudes and frequency of the external
gential velocity (dashed line) and the external vorticity perturba-
perturbations,the structure of the perturbed flow inside tions (lines for ξ > 1) in the case of the shell-like flow (l = 10,
the shock wave is determined by solving in the interval s = 0.1304i, ω = 2.7, γ = 4/3). The imaginary parts are identi-
ξ ξ 1, ξ 0 the system of linear ODEs (21)- callyzero.
in in
≤ ≤ ≤
(23),(25) for unknown variables f (ξ), f (ξ), f (ξ),
vρ vξ vτ All these coefficients of amplifications are ratios of am-
fp(ξ) with the outer (35)-(37) and inner (52), (59) or plitudes and time-constant by definition.
(60)boundaryconditions. Theseunknownvariablesrep- First let us consider the vortex mode whose frequency
resent the eigenfunctions for the problem while η is now is given by Equation (33).
an eigenvalue. We find the root η by multiple shoot- Thesimplestperturbationofthevelocityfieldisauni-
ingmethodterminatingiterationsuponreachingrelative form motion δv = const which contains only a dipole
precisionatleast10−4. Inthecaseofthefilledcavitythe component in sp0herical harmonics expansion and corre-
problemmaybecomestiff;thisdictatessometimesaneed spondstoasteadydriftofthewholeshockwave. Sucha
for much better accuracy to satisfy the inner boundary flow is purely irrotational, the denominator in (68) van-
condition. Figure 3 demonstrates the typical behavior ishes but the numerator remains finite. We can observe
of the eigenfunctions in the case of shell-like flow. this effect as a resonance at l = 1 in the case of the
Toestimatethepossibleamplificationofperturbations uniform background (Figure 4). The same effect exists
bytheshockwaveweintroducethefollowingcoefficients of course in the case of any background which involves
of δρ = 0 however, but since we consider the vortex and
0
density amplification thee6ntropymodesseparately,wedonotobservethisres-
onance in other figures.
f
R= | ρs|, (66) Let us remember that the curl of velocity for any har-
f
| ρ0| monic (20) is expressed as a product of an amplitude
velocity amplification depending on ξ and the angular part lYlm. Thus the
coefficient(68)actually presents the amplitudes ratio ir-
f
Ξ= | vs|, (67) respectiveoflYlm. Thelimitl 0thusprovidesaneval-
f uationofindeterminate forma→szerodivide zerobecause
| v0| the vorticitiesthemselvesδv (l =0)andδv (l =0)van-
and vorticity amplification 0 s
ish.
curlf FromFigures 4,5 we see that vorticityintensifies sub-
Ω= | vs|. (68)
curlf stantially up to 20 times within the wavenumbers range
| v0| under consideration out of resonance while the veloc-
Accountingforanexplicitformforthe curlofthe vector ity fluctuations are damped. The reason for vorticity
δv withzerobinormalcomponentinnaturalcoordinates amplification is that the rippled shock front generates
the shear flow. While the radial velocity perturbations
dδv δv δv
δv=e τ + τ ξ , (69) are strongly suppressed the tangential perturbations are
b
∇× dξ ξ − ξ
(cid:18) (cid:19) excited. For instance, for the shortwave perturbation
wecanwriteoutthe vorticityamplificationcoefficientin l=10whoseeigenfunctionsaredepictedinFigure3,the
full as preshock velocity perturbations are radially dominated
Ω= |ξdfdvξτ s+fvτs −fvξs|. (70) damuepltioturdeelsatoiofnth(e28p)os(tfsvhξo0c=k v1e.l0o,cfitvyτ0co=m0p.o0n0e6n)tsbuetqutahle-
f f
|(cid:12)(cid:12)vξ0 − vτ0|
Resonant Amplification of Turbulence by Blast Waves 9
40 1
(a) (a)
0.9
35
γ = 1.1 0.8
30 ω = 0. γ = 1.1
s = 0.6i 0.7 ω = 0.
25 s = 0.6i
0.6
Ω 20 Ξ0.5
0.4
15
0.3
10
0.2
5
0.1
0 0
10−2 10−1 100 101 102 10−2 10−1 100 101 102
l l
10 1.5
(b) (b)
9
γ = 4/3 γ = 4/3
8
ω = 2.7 ω = 2.7
7 s ≈ 0.1304i s ≈ 0.1304i
1
6
Ω 5 Ξ
4
0.5
2.8 0.5
3 0.4
2 2.7 0.3
2.6 0.2
1
17 18 19 17.95 18 18.05
0 0
10−2 10−1 100 101 102 10−2 10−1 100 101 102
l l
Figure 4. The vorticity amplification coefficient (Eq. (68)) as a Figure 5. ThesameasinFig.4butforthevelocityamplification
functionoftheorbitalwavenumber lfortheuniform(upper plot) (Eq.(67)).
andthenonuniformbackground(lowerplot).
Resonance prescribedby Equation (62) at l=0, ω =
ize (fvξs = 0.19,fvτs = 0.17). Along with that the tan- 0 has a greater width: ∆l =2. This resonance describes
gentialperturbationderivativedf /dξundergoesasteep self-excitation of radial fluctuations of the shock at neg-
vτ
rise from zero before the shock to 3.1 behind the shock ligibleexternaldensityperturbationsandtakesplacefor
(seerightbottominsetinFigure3)building upvorticity arbitraryωbecausedispersiondependenciesalwayscom-
significantly (see Equation (70)). prise the point (s = 0, l = 0). Suppose the energy of
The most intriguing effect is the resonance that we explosionundergoes small perturbation E E +δE .
0 0 0
→
find in the case ω = 2.7 at the resonant wavenumber Thentherelativedisplacementoftheshockfrontchanges
l=l∗ givenby Equation(65). The insets in Figures 4,5 over time as
showthattheresonancewidthisrathersmallandequals
R (E +δE ,t) R (E ,t)
s 0 0 s 0
∆l 0.4 for vorticity and < 0.1 for velocity amplifica- η = −
≈ R (E ,t)
tion. Here the resonance width is defined as a range s 0
tohfew‘amveeannu’mvbaelurseslbiyntwhehivcahluteheoffr∆eqImuesncy0d.i1ff.erSstrfircotmly = (E0+δE0)5−1ωt5−2ω −E05−1ωt5−2ω t0, (71)
speaking, according to the linear theory∼of oscillations E5−1ωt5−2ω ∝
0
the width of resonance in the dissipationless case is neg-
ligible so long as the width is usually defined as the in- that is, according to Equation (39) the spherical pertur-
terval at which the frequency differs no more than twice bation will have zero frequency.
from the maximum value.
Thedensityperturbationsintheentropymodeaream- 4. DISCUSSION ANDCONCLUSION
plified out of resonance only in the case of the accelerat- Our linear analysis validates an idea that the blast
ing shell blast wave (ω =2.7) whereas they are damped wave can amplify both density and vorticity perturba-
by decelerating solid blast wave (ω =0) (Figure 6). tionsintheturbulizedinterstellarmediumthroughreso-
10 Zankovich and Kovalenko
×103 dissipation increase. The linear theory of oscillations
10 predicts that the resonance width of frequencies is pro-
(a) portional to the doubled decay coefficient. Taking
9
into account that in our self-similar problem the fre-
8 γ = 1.1 quency is defined as slog(t/t0)/t and the decay has to
be stipulated by the turbulent viscosity we find that
7 ω = 0. ∆Imslog(t/t )/t 2ν /λ2, where ν is the tur-
0 turb turb
s = 0. ≈
bulent kinematic viscosity. The length of the pertur-
6
bation inside the shock wave can be estimated as λ
5 2πR /√l2+n2 while the kinematic viscosity approx≈i-
R s
mately is ν 1l c , where c is the postshock
4 turb ≈ 3 turb s s
sound velocity and l is the characteristic scale of a
turb
3 turbulentvortex. Expressingtheresonantwidthinterms
of the wavenumbers for the resonant case considered in
2 our model, n=0 and l =l∗, we get
1 ∗ −1
∆l l l c t dIms
turb s
. (72)
0 l∗ ≈ 6π2R2log(t/t ) dl
10−1 100 101 102 s 0 (cid:18) (cid:12)l∗(cid:19)
(cid:12)
l The postshock soundvelocity is determ(cid:12)ined through the
(cid:12)
relations (10),(12):
7.6
R
s
(b) c =Z , (73)
s
t
7.5
where
2 2γ(γ 1)
Z = − . (74)
7.4 (5 ω)(γ+1)
p−
γ = 4/3
For γ = 1.1 and ω = 0 we have Z 0.089, for γ =
ω = 2.7 ≈
7.3 4/3, ω = 2.7 the coefficient Z is something like 0.35.
R
s ≈ 2.3478i
Since the propagatingshock embraces growingscales we
estimate the turbulent vortex scale l as proportional
turb
7.2
to the shock front radius R with a fixed constant of
s
proportionality ξ . Collecting all estimates we finally
turb
obtain
7.1
∗ −1
∆l l ξ Z dIms
turb
. (75)
170−2 10−1 100 101 102 l∗ ≈ 6π2log(t/t0)(cid:18) dl (cid:12)(cid:12)l∗(cid:19)
l Thenumericalcalculations(Fig.2)sho(cid:12)wthatthederiva-
(cid:12)
tive of the dispersion relation dIms amounts to 0.1
Figure 6. ThesameasinFig.4butforthedensityamplification ∗ dl l∗ ∼
(Eq.(66)). at the resonant point l = 18. Assu(cid:12)ming ξturb ≈ 0.1 we
find (cid:12)
nant shock-turbulence interaction. This result is consis- ∆l
log(t/t ) 0.025, (76)
tentwithanalyticandnumericalstudies(Leeetal.1997; l∗ 0 ≈
Jamme et al. 2002;Wouchuk et al. 2009)predicting vor-
which is close to the width of resonance found numeri-
ticity amplification by a distorted shock front. As dis-
cally (Figs. 4b,5b).
tinct from the aforementioned works we study the non-
The estimates adduced show that the resonancerange
local mechanism of amplification when the wave lengths
subjecttotheexistenceofdissipationissmallbutwidens
λ of perturbations are not negligible compared with the
andcan be not vanishingly smallif the resonanceoccurs
shockwavesizeR . This singlesoutthe resonantnondi-
s at a more gently sloping dispersion curve (dIms 0)
mensional wavenumbers at macro- (l = 2πRs/λ ∼ 1), and for higher order harmonics l∗ and n. dl l∗ →
meso- (l 18) and microscopic (l > 200) scales. Res-
(cid:12)
onant me≈soscales are surprisingly very close to the ob- Secondly,toexcite resonantoscillationsnotice(cid:12)ably the
amplitude of resonant harmonics in the spectrum of ex-
served scales (l 15) of corrugation structures in the
≈ ternal perturbations should be non-negligible. Suppose
remnant SNR 0509-67.5.
turbulent fluctuations energy distribution over discrete
On the other hand, there are some factors negating
orbital quantum numbers l has a power dependence like
the effectiveness of the resonant mechanism. First, the
resonancewidth is small (typically ∆l 1) and the res- Kolmogorov law f2 l−5/3, and if we are fortunate it
onancepointcanbeclampedbetweent≪wonearlyinteger startsfromlmin whler∼elmin isclosetotheresonantnum-
∗
points l. Since the observedorbitalwavenumbersl must ber l =18, then the relative square of amplitude of the
∞
be integer values, the amplification for these will not be l = lmin = 18 mode amounts to fl2min/( l=lminfl2) ≈
too large. 3.7% of the total energy of fluctuations. The estimate
∗ P
The resonance range can grow in size however with can be reduced significantly if l <l .
min