Stability of Vertically and Radially Stratified Protoplanetary Disks

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1 Stability of Vertically and Radially Stratified Protoplanetary Disks Figure from Stoll & Kley (2014) Glen Stewart Laboratory for Atmospheric and Space Physics University of Colorado

2 Vertical Shear instability simulated by: Nelson et al. (MNRAS 435, , 2013) Richard et al. (MNRAS 456, , 2016) A radial entropy gradient produces vertical shear in the angular velocity of protoplanetary disks. This can lead to a convective instability in vertically stratified disks even when the vertical entropy gradient is stable. (Urpin and Brandenburg 1998) Nelson et al. (2013)

3 Motivation Why is a protoplanetary disk with radial and vertical temperature gradients unstable even if the vertical lapse rate is stable? What is the most simple model that will include the vertical shear instability simulated by Nelson et al. (2013)? Our solution: Use an axisymmetric anelastic disk model that filters out pressure waves. This is justified by the slow convective velocities that are << the sound speed in the simulations. (see Knobloch and Spruit (1986) for similar approach) Main advantage is that the dynamics are formally similar to the symmetric baroclinic instability in planetary atmospheres! (see Holton s text: An Introduction to Dynamic Meteorology)

4 Vertical component of the Sun s gravity vanishes at the disk midplane (z = 0). z r 0 x F = - GM r z» - GM 1- z 2 r 0 2 2r 0 2 ö F z = - F = - GM z r z 2 ( )» - GM r 0 z = -W 2 K z W K = angular velocity of an orbit

5 Linearize about axisymmetric flow density = r ( r, z), radial velocity = 0 + u( r,z,t), angular momentum = J(r,z) + j( r,z,t), vertical velocity = 0 + w( r,z,t), entropy = S(r,z) + s( r,z,t). Unperturbed state is specified by the three functions r( r,z), J ( r,z), S( r,z), that satisfy 1 P r + F = 0, 1 P r r + F r = J 2 r, 3 P = K ( r)r G ( r,z), S = C p ln P1 g ö r.

6 Radial & Vertical Stratification in a Polytropic Disk Density profile: é ê ê ë r( r,z) = r 0 ( r) 1+ ( G -1) h 2 ( r) r( r,0) = r 0 ( r) µ r p, T ( r,0) = T 0 ( r) µ r q, ( ) 1 G z 2 r -1 ö ù ú 2 ú û Aspect ratio of disk: h( r) = G P r = H r W K r µ r ( 1+q ) 2 Angular Momentum profile: J 2 ( r,z) r 4 2 W K r G h2 r ( ) = 1+ p + q ( ) + q + p ( 1- G ) G z 2 r 2 ö Typical exponents are p = -1.5, q = -1, G =1.01

7 Radial & Vertical Entropy Gradient S r = C p S = C p é ê ë q + p( 1- G) g r G g -1 ö ln r = - C p 2 c s + G g -1 ö ln r ù ú, r û F G -g g. Vertical buoyancy frequency: N z 2 = 1 C p F S» g - G ö g z 2 H 2 ö W 2 K In the adiabatic case, g = G, get a thermal wind balance: 1 J 2 r 3 = F 1 S C p r

8 Linearized anelastic equations of motion r rru ( ) + rrw ( ) = 0, u t - 2W j r = - r d P ö r, j t + u J r + w J = 0, w t = - d P ö r + F s t + u S r + w S = - s t. s C p, (radiative damping) These equations can be reduced to just three equations by defining a stream function: f f rru = -, rrw = r.

9 Anelastic Evolution Equations Stream function/vorticity: ( f,g) º 1 r f r g - f g r ö z = u - w r = - r 1 f ö rr r - 1 f ö rr Vorticity: Angular Momentum: Entropy: t + f,z ö r = J + j,w Kr 2 j t + 1 r ( f, J + j ) = 0 s t + 1 r f,s + s ( ) = - s t ( ) + S + s ( ) C p,rf r 0,z ö Structure is similar to the Symmetric Baroclinic instability!

10 Solberg-Hoiland Stability Criteria 1 r 3 J 2 r - 1 rc p ÑP iñs > 0, F J 2 r S - J 2 S r ö > 0. Historically, this criteria was derived using parcel displacements. They can also be derived using Hamiltonian methods to derive the available energy (Codoban and Shepherd 2006). For circumstellar disks, these criteria are usually satisfied, so that radiative damping must be added to get an instability. This was already discovered for rotating stars in the 1960 s by Goldreich and Schubert and independently by Fricke. For disks, this idea was derived by Urpin and Brandenberg (1998) and later discovered numerically by Nelson, Gressel and Umurhan (2013).

11 Linear Stability Analysis Since the radial stratification is weak, we can use a WKB approximation and expand variables using Fourier series in the radial direction. Since the instability can span the entire scale height of the disk, we want to keep the full z-dependence of the unperturbed state. We therefore use Chebyshev collocation methods to resolve the vertical structure. The result is a generalized numerical eigenvalue problem.

12 Numerical stability results using compressible model (Lin and Youdin 2015) Numerical stability results using anelastic model. (Stewart 2016)

13 Fundamental mode causes corrugations in the disk midplane (Lin and Youdin 2015) Note that the horizontal scale has been expanded by a factor or ten!

14 Conclusions Convective instability occurs for realistic radial entropy gradients, even when the vertical entropy gradient would imply stability. A large thermal diffusivity (or numerical viscosity) can kill the instability. This is why most simulations do not find this instability. The instability is most vigorous for rapid cooling. For longer cooling times, the instability is less effective because the growth rate slows and shifts to smaller scales, which are subject to turbulent decay. Growth of the instability is suppressed by long cooling times in the opaque inner disk and in the optically thin outer disk. Is their a sweet spot in the MRI dead zone of disks where planets form? But what about magnetic fields in the upper layers of the disk?

15 Adding the Hall Effect to Disk Simulations Hall effect amplifies the magnetic field above z = 2H! (Bai 2014)

16 Modified Solberg-Hoiland Stability Criteria r W2 r - 1 rc p ÑP iñs > 0, F W 2 r S - W2 Balbus (2001) finds that a magnetic field can fundamentally modify the stability criteria for stratified flows such that the angular velocity replaces the angular momentum! A similar replacement occurs in the magnetorotational instability for unstratified disks. S r ö > 0. Does this imply that the upper layers of a disk are always unstable to convective motions? How does the Hall effect modify Balbus stability criteria? Can they be derived using Hamiltonian methods? (work in progress!)

17 The vertical shear instability may trigger subcritical baroclinic instabilites in the disk, leading to vortices! (Richard et al. 2016)