Baroclinic annulus flows with internal heating
jump to navigationThe use of direct internal heating of the fluid, in addition to conductive heating/cooling at the cylindrical side boundaries, allows greater flexibility in exploring the influence of different background thermal structure on baroclinic instability. Experimentally, internal heating may be applied by using a weak electrolyte (a dilute solution of an inorganic salt) as the working fluid and applying an alternating electric current between the inner and outer sidewall. A schematic arrangement is shown below.
Because of the cylindrical curvature, this arrangement leads to a spatially-varying heat source term with a strength proportional to 1/r2 - so more heating per unit volume appears near the inner cylinder than further out.
Background flow patterns
Since the working fluid is now heated in the interior of the annular channel, we are free to apply cooling, either at one or both sidewalls. Where we cool at just one boundary (the inner or outer cylinder), the other may be thermally insulated (so dT/dr = 0). By the thermal wind equation, dT/dr = 0 implies that dv/dz = 0. Taken together with the Ekman boundary condition at the bottom of the tank, this means that the mean azimuthal velocity v itself is likely to be zero wherever dT/dr = 0. By choosing different combinations of internal heating and/or sidewall cooling at either or both cylinder, therefore, we can set up a range of different azimuthal flow patterns, all with anticyclonic shear at upper levels, but with flow which is either monotonic with radius or reverses in a pair of opposing jets. The possible flow combinations are illustrated below.
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Use of an insulating boundary at either the inner or outer sidewall leads to a strongly sheared azimuthal flow tending to zero at the insulated boundary. Cooling at both sidewalls, however, leads to dT/dr = 0 occurring in mid-channel. Consistency with thermal wind balance leads to an anticyclonic shear zone at upper levels spanning the region between two opposing baroclinic jets - prograde near the inner cylinder and retrograde near the outer.
When baroclinic waves develop in such a system, their superposition leads to wave patterns of very different appearance, depending upon where the wavy flow reinforces or opposes the basic thermally-driven azimuthal flow:
- Wavy flow near conducting sidewalls reinforces the basic azimuthal flow in the anticyclonic part of the wave (at upper levels - the opposite happens at lower levels where the cyclonic parts of the wave reinforce the cyclonic shear of the background flow).
- Wavy flow near insulating sidewalls tends to dominate or even oppose the background flow, leading to azimuthal flows which alternately flow prograde or retrograde parallel to the boundary. This can lead to the appearance of a meandering jet stream enclosing closed, recirculating vortices adjacent to the insulated boundary.
- Wavy flow in a system with cooling at both sidewalls leads to a pattern where the cyclonic parts of the wave pattern oppose the shear of the background azimuthal flow, whereas the anticyclonic parts reinforce the background flow (with the opposite effect at lower levels). This leads to chains of closed, anticyclonic vortices at upper levels, apparently separated by relatively quiescent regions in between.
Flow regimes
Like their more classical boundary-only heated/cooled counterparts, baroclinic waves with internal heating exhibit a wide range of different types of flow regime, depending upon the strength of heating and background rotation rate Ω. We can map the behaviour of the flow on a regime diagram as illustrated below (obtained for an internally heated flow with cooling at both sidewalls by Read et al. 1997).
- Axisymmetric flow occurs at low values of Ω for a given heating rate
- Regular, coherent waves occur at moderate values of Ω, with a tendency for higher azimuthal wavenumbers at larger Ω and Taylor number.
- Irregular, more chaotic flows occur at high Ω and large Taylor number/small Hide number.