Scaling of co nvec tio n a n d pl a t e t ec t o nic s in Su pe r - Eart hs

The di scovery of t h r e e Supe r - Eart h s a r o u n d diffe re n t s t a r s, p o s si ble o nly in t h e las t yea r, p r o m p t s u s t o s t u dy t h e c ha r ac t e ri s tics of o u r pl a n e t wit hi n a ge n e r al co n t ex t. The Ear t h, b ei ng t h e m o s t m a s sive t e r r e s t rial o bject in t h e s ol a r sy s t e m is t h e o nly pl a n e t t h a t ex hi bi t s pl a t e t ec t o nics. We t hi n k t hi s mig h t n o t b e a coi nci de nce a n d ex plo r e t h e r ole t h a t m a s s pl ay s in d e t e r mi ni ng t h e m o d e of co nvectio n. We u s e t h e sc ali ng of co nvective vigor wit h Rayleigh n u m b e r co m m o nly u s e d in p a r a me t e ri z e d co nvec ti o n. We s t u dy h o w t h e p a r a me t e r s co n t r olli ng co nvec tio n: Rayleig h n u m b e r (Ra), b o u n d a ry layer t hick ne s s (del t a), in t e r n al t e m p e r a t u r e (Ti) a n d co nvec tive velocities (u) s cale wit h m a s s. Thi s is p o s si ble f r o m t h e sc ali ng of h e a t fl ux, m a n tl e d e n si ty, si z e a n d gr avity wit h m a s s whic h we r e p o r t e d in Vale ncia, e t. al 2 0 0 6. The ext r a p ol a ti o n t o m a s sive r ocky pl a n e t s is d o n e f r o m o u r k n owle dge of t h e Ear t h. Even t h o ug h u n ce r t ai n tie s a ri se fr o m ext r a p ol a ti o n a n d a s s u m p ti o n s a r e n e e d e d we co n si d e r t hi s si m ple sc ali ng t o b e a fir s t a d e q u a t e s t e p. As t h e m a s s of a pl a n e t incr ea s e s, Ra inc rea s e s, yiel di ng a d ec r e a s e in d el t a a n d a n incre a s e in u, while Ti incr ea s e s very slig h tly. Thi s is t r u e for a n isovi sco u s cas e a n d m o r e acce n t u a t e d in a t e m p e r a t u r e d e p e n d e n t visco sity s ce n a rio. In a pl a n e t wi t h vigor o u s co nvectio n (high u), a t hi n lit h o s p h e r e (low d el t a) is ea sier t o s u b d uc t a n d h e nce, initia t e pl a t e t ect o nics . The lit h o s p h e r e al s o h a s t o b e d e n s e e n o u g h (col d a n d t hick) t o h ave t h e b o uya ncy n ece s s a ry fo r s u b d u c ti o n. We calc ula t e t h a t a co nvective cycle fo r a n is ovisco u s pl a n e t is \ t a u ~M^( - 0. 3) co n si d e ri ng wh ole m a n tle co nvectio n. Mea ni ng t h a t if t h e s e pl a n e t s h ave co n ti ne n t s, t h e ti me scale for co n ti ne n t al r e a r r a n ge me n t is s h o r t e r (abo u t h alf t h e Ear t h' s fo r a 5 e a r t h - m a s s pl a n e t). Ad di ti o n ally, we ex pl or e t h e n ega tive fee d b ack cycle b e t wee n co nvec tio n a n d t e m p e r a t u r e d e p e n d e n t visco sity a n d e s ti ma t e a ti me sc ale fo r t hi s effect.