Slowly-Growing Spiral Mode Instabilities in Protostellar Disks · exponential spiral mode growth in...
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Slowly-Growing Spiral Mode Instabilities in Protostellar Disks Introduction Linear Mode Analysis Benisty et al. 2015 Wagner et al. 2018 Over the past several decades, computational fluid dynamics has advanced rapidly, and the range of available numerical algorithms and computationally feasible physical problems has expanded. Modern numerical solvers provide a compelling opportunity to probe for as-yet undiscovered effects that emerge with longer integrations and higher numerical precision. Improved resolution permits identification of slowly- growing instabilities which may have a close connection to structures observed by ALMA and SPHERE. ℒ "# $ % =0. In this work, we study eigenvalues corresponding to exponential spiral mode growth in the linearized equations of hydrodynamics. By generalizing the procedure outlined in Adams et al. 1989 and applying this to various disk density profiles, we gain further insight into the linear evolution of disks. We use our linear solutions to the hydrodynamic governing equations to assess the fidelity of meshless and conventional grid-based hydrodynamic solvers. Next, we will use our results to study the weakly nonlinear long-term development of endogenously developed spiral modes. We will also explore how modern fluid codes maintain recently discovered eccentric disk mode solutions that display unperturbed structure at the inner and outer disk boundaries (Lee et al. in prep). By comparing modern simulations with prior results, we hope to provide a stronger understanding of the impact of fluid mechanics upon the evolution of protostellar disks. x + i = 0 Malena Rice 1,2 & Gregory Laughlin 1 1 Yale University – Department of Astronomy, 2 NSF Graduate Research Fellow Background & Theory Directly imaged observations of spiral structures in protoplanetary disks studied by VLT/SPHERE. Continued Work ( ) − +Ω - . - / + . 1 + . - / 2- / 21 3 1 + 25 . 21 − "6 1 7 1 =0 ( ) − +Ω 3 1 − 2Ω7 1 =− 2 21 (: 1 +ℎ 1 ) ( ) − +Ω 7 1 + = > >? 3 1 = "6 1 (: 1 +ℎ 1 ) : 1 = −2@A B / ∞ C + D,F G 1 F HF For a given density profile, the linear equations of hydrodynamics can be rearranged into a homogeneous integro-differential equation which can then be solved to find eigenvalues ) corresponding to exponential spiral mode growth. This can be expressed as a complex matrix which we denote as ℒ "# , with eigenvector $ % . For a power law density profile G D =G ∗ ( J ∗ K ) L , we obtain: ℒ "# = {N "O > + PD + 2 1−L −1N "O . +Q R SD > + PD 1−L + L L − 1 ]U "O V O# + W XK {N "# > + PD − 2Y − 1 N "# . + [SD > − PDY + Y Y + 1 ]U "# }− \ ] W^_ ] K ]`ab c U "# +U .6 (D > S + DP) W ] ) > K d a(e ∗ fe g ) h "# .
Transcript of Slowly-Growing Spiral Mode Instabilities in Protostellar Disks · exponential spiral mode growth in...
Slowly-Growing Spiral Mode Instabilities in Protostellar Disks
Introduction Linear Mode Analysis
Benisty et al. 2015Wagner et al. 2018
Over the past several decades, computational fluid dynamics has advanced rapidly, and the range of available numerical algorithms and computationally feasible physical problems has expanded. Modern numerical solvers provide a compelling opportunity to probe for as-yet undiscovered effects that emerge with longer integrations and higher numerical precision. Improved resolution permits identification of slowly-growing instabilities which may have a close connection to structures observed by ALMA and SPHERE.
ℒ"#$% = 0.
In this work, we study eigenvalues corresponding to exponential spiral mode growth in the linearized equations of hydrodynamics. By generalizing the procedure outlined in Adams et al. 1989 and applying this to various disk density profiles, we gain further insight into the linear evolution of disks. We use our linear solutions to the hydrodynamic governing equations to assess the fidelity of meshless and conventional grid-based hydrodynamic solvers.
Next, we will use our results to study the weakly nonlinear long-term development of endogenously developed spiral modes. We will also explore how modern fluid codes maintain recently discovered eccentric disk mode solutions that display unperturbed structure at the inner and outer disk boundaries (Lee et al. in prep). By comparing modern simulations with prior results, we hope to provide a stronger understanding of the impact of fluid mechanics upon the evolution of protostellar disks.
x+ i = 0
Malena Rice1,2 & Gregory Laughlin11Yale University – Department of Astronomy, 2NSF Graduate Research Fellow
Background & Theory
Directly imaged observations ofspiral structures in protoplanetarydisks studied by VLT/SPHERE.
Continued Work
( ) −+Ω -.-/+ .
1 +.-/2-/21 31 + 25.
21 −"61 71 = 0
( ) −+Ω 31 − 2Ω71 = − 221(:1 + ℎ1)
( ) −+Ω 71 + =>>?31 =
"61 (:1 + ℎ1)
:1 = −2@AB/
∞C+ D, F G1 F HF
For a given density profile, the linear equations of hydrodynamics can be rearranged into a homogeneous integro-differential equation which can then be solved to find eigenvalues ) corresponding to exponential spiral mode growth. This can be expressed as a complex matrix which we denote as ℒ"#, with eigenvector $%.
For a power law density profile G D = G∗(J∗K )L, we obtain:
ℒ"# = {N"O> + PD + 2 1 − L − 1 N"O
. + QR
SD> + PD 1 − L +L L − 1 ]U"O VO# + W
XK{N"#> + PD − 2Y − 1 N"#
. + [SD> − PDY +Y Y + 1 ]U"#} − \] W^_] K
]`abcU"# + U.6(D>S + DP)W])
> Kda(e∗feg)
h"#.
