Inertial waves in a differentially rotating spherical shell
May 30, 2012 - 1. ): the wave's Doppler-shifted frequency the (uniform) angular frequency profile ... existence of critical latitudes . damping rate: scaling in E. 1/3.
Base flow: Rotation profile: cylindrical – Ω0(s=rsinθ) – or shellular – Ω0(r).
ηR
Methods (i) Linearized equations + stress-free B.C. solved with a spectral code (ii) Inviscid case + WKB approximation: method of characteristics
Inertial modes with solid-body rotation Linearized governing equations: with the (uniform) angular frequency profile
Look for solutions in
Inertial modes with solid-body rotation Linearized governing equations: with the (uniform) angular frequency profile
Look for solutions in Poincaré equation (inviscid fluid + WKB; p ≡ h1): where
the wave's Doppler-shifted frequency
Inertial modes with solid-body rotation Linearized governing equations: with the (uniform) angular frequency profile
Look for solutions in Poincaré equation: where
→ paths of characteristics:
the wave's Doppler-shifted frequency
with z
Inertial waves propagate for and follow straight paths of characteristics in a meridional slice of the shell →s
Inertial modes with cylindrical rotation Linearized governing equations: with
Cylindrical rotation profile:
with
(Proudman-Taylor)
Inertial modes with cylindrical rotation Linearized governing equations: with
Cylindrical rotation profile:
with
(Proudman-Taylor)
Poincaré equation: and
where
→ paths of characteristics:
with
. ξ(s) → differential rotation bends paths of characteristics, . ξ(s) > 0 → inertial waves may propagate: - in the whole shell: D modes (ξ > 0 for all s), - or in only part of it: DT modes (existence of Turning surfaces where ξ(s)=0 )
Differential rotation parameter
Cylindrical rotation: m=0 eigenmodes
|eigenfrequency| in the inertial frame
D modes: hyperbolic within the whole shell
DT modes: hyperbolic in part of the shell (existence of Turning surfaces)
Cylindrical rotation: m=0 eigenmodes
Example of m=0 D mode with a wave attractor
+ ω(s) = 1 - 0.3s2 kinetic energy
Dynamics of characteristics (inviscid, WKB)
Viscous eigenvalue problem
Cylindrical rotation: m=0 eigenmodes
Example of m=0 DT mode with two wave attractors
+ ω(s) = 1 - 0.4s2
turning surface
kinetic energy
Dynamics of characteristics (inviscid, WKB)
Viscous eigenvalue problem
Cylindrical rotation: m=0 eigenmodes
kinetic energy
turning surface
+ Example of m=0 DT mode with a wave attractor with no inner core
Viscous eigenvalue problem
Differential rotation parameter
Cylindrical rotation: m≠0 eigenmodes
Existence of critical 'cylinders', where ώp(s) = 0 (=corotation resonance)
. phase velocity → 0 . group velocity → 0 . characteristics → parallel to the rotation axis
eigenfrequency in the inertial frame
Cylindrical rotation: m≠0 eigenmodes
+
Example of m=2 D mode with a critical cylinder (ώp = 0):
critical 'cylinder'
kinetic energy
→ strong absorption at corotation Viscous eigenvalue problem
Inertial waves with shellular rotation Very similar results with 'shellular' rotation profile ω(r) = rσ . paths of characteristics obey a 2nd order PDE of mixed type . shear layers ↔ characteristics kinetic energy
turning surface
. existence of critical latitudes → sin θ = ώp(r) / 2ω(r)
Viscous eigenvalue problem
Inertial waves with shellular rotation Very similar results with 'shellular' rotation profile ω(r) = rσ . paths of characteristics obey a 2nd order PDE of mixed type . shear layers ↔ characteristics . existence of critical latitudes . damping rate: scaling in E1/3 for most modes (E: Ekman number)
τ indpt of E! turning surface
m=0 DT modes
Inertial waves with shellular rotation … but very different behavior at critical shells, where ώp(r) = 0 – ω(r) = rσ – At corotation: . phase velocity → 0 (as for cylindrical and solid-body rotation) . group velocity may → 0 OR → ∞: waves may cross corotation
critical shell
Viscous eigenvalue problem
Inertial waves with shellular rotation … but very different behavior at critical shells, where ώp(r) = 0 – ω(r) = rσ – At corotation: . phase velocity → 0 (as for cylindrical and solid-body rotation) . group velocity may → 0 OR → ∞: waves may cross corotation . modes with a corotation resonance may be unstable at small enough Ekman numbers
unstable
Inertial waves with differential rotation: summary Baruteau & Rieutord, submitted to Journal of Fluid Mechanics arxiv.org/abs/1203.4347
Propagation properties of linear inertial waves in a rotating fluid contained in a spherical shell. Astrophysical and fluid lab. applications.
With differential rotation: - Singular modes still exist with thin shear layers following short-period wave attractors. They may exist even in the full sphere. - Possible existence of turning surfaces within the shell. - Most modes have a damping rate in E1/3; some in E0. - Preliminary study of the behavior at a corotation resonance: . cylindrical rotation: waves absorbed at corotation, . shellular rotation: waves may propagate across corotation with no visible absorption. Corresponding modes unstable at small E.
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