### Abstract

The effect of rotation on horizontal convection in a cylindrical enclosure is
investigated numerically. The thermal forcing is applied radially on the bottom
boundary from the coincident axes of rotation and geometric symmetry of the
enclosure. First, a spectral element method is used to obtain axisymmetric basic flow
solutions to the time-dependent incompressible Navier–Stokes equations coupled via a
Boussinesq approximation to a thermal transport equation for temperature. Solutions
are obtained primarily at Rayleigh number *Ra* = 10^{9}
and rotation parameters up to
*Q =* 60 (where Q is a non-dimensional ratio between thermal boundary layer thickness
and Ekman layer depth) at a fixed Prandtl number *Pr* = 6.14 representative of water
and enclosure height-to-radius ratio *H/R* = 0.4. The axisymmetric solutions are
consistently steady state at these parameters, and transition from a regime unaffected
by rotation to an intermediate regime occurs at *Q* ≈ 1 in which variation in thermal
boundary layer thickness and Nusselt number are shown to be governed by a scaling
proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an
increase in *Q* sees the flow accumulate available potential energy and more strongly
satisfy an inviscid change in potential energy criterion for baroclinic instability.
At the strongest *Q* the flow is dominated by rotation, accumulation of available
potential energy ceases and horizontal convection is suppressed. A linear stability
analysis reveals several instability mode branches, with dominant wavenumbers
typically scaling with *Q*. Analysis of contributing terms of an azimuthally averaged
perturbation kinetic energy equation applied to instability eigenmodes reveals that
energy production by shear in the axisymmetric mean flow is negligible relative
to that produced by conversion of available potential energy from the mean flow.
An evolution equation for the quantity that facilitates this exchange, the vertical
advective buoyancy flux, reveals that a baroclinic instability mechanism dominates
over 5≲Q≲30, whereas stronger and weaker rotations are destabilised by vertical
thermal gradients in the mean flow.

Original language | English |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Journal of Fluid Mechanics |

Volume | 795 |

DOIs | |

Publication status | Published - May 2016 |

### Keywords

- Convection
- Instability
- Rotating flows

### Cite this

*Journal of Fluid Mechanics*,

*795*, 1-35. https://doi.org/10.1017/jfm.2016.193

}

*Journal of Fluid Mechanics*, vol. 795, pp. 1-35. https://doi.org/10.1017/jfm.2016.193

**Linear stability and energetics of rotating radial horizontal convection.** / Sheard, Gregory; Hussam, Wisam K.; Tsai, TzeKih.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Linear stability and energetics of rotating radial horizontal convection

AU - Sheard, Gregory

AU - Hussam, Wisam K.

AU - Tsai, TzeKih

PY - 2016/5

Y1 - 2016/5

N2 - The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number Ra = 109 and rotation parameters up to Q = 60 (where Q is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number Pr = 6.14 representative of water and enclosure height-to-radius ratio H/R = 0.4. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at Q ≈ 1 in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in Q sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest Q the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with Q. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over 5≲Q≲30, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

AB - The effect of rotation on horizontal convection in a cylindrical enclosure is investigated numerically. The thermal forcing is applied radially on the bottom boundary from the coincident axes of rotation and geometric symmetry of the enclosure. First, a spectral element method is used to obtain axisymmetric basic flow solutions to the time-dependent incompressible Navier–Stokes equations coupled via a Boussinesq approximation to a thermal transport equation for temperature. Solutions are obtained primarily at Rayleigh number Ra = 109 and rotation parameters up to Q = 60 (where Q is a non-dimensional ratio between thermal boundary layer thickness and Ekman layer depth) at a fixed Prandtl number Pr = 6.14 representative of water and enclosure height-to-radius ratio H/R = 0.4. The axisymmetric solutions are consistently steady state at these parameters, and transition from a regime unaffected by rotation to an intermediate regime occurs at Q ≈ 1 in which variation in thermal boundary layer thickness and Nusselt number are shown to be governed by a scaling proposed by Stern (1975, Ocean Circulation Physics. Academic). In this regime an increase in Q sees the flow accumulate available potential energy and more strongly satisfy an inviscid change in potential energy criterion for baroclinic instability. At the strongest Q the flow is dominated by rotation, accumulation of available potential energy ceases and horizontal convection is suppressed. A linear stability analysis reveals several instability mode branches, with dominant wavenumbers typically scaling with Q. Analysis of contributing terms of an azimuthally averaged perturbation kinetic energy equation applied to instability eigenmodes reveals that energy production by shear in the axisymmetric mean flow is negligible relative to that produced by conversion of available potential energy from the mean flow. An evolution equation for the quantity that facilitates this exchange, the vertical advective buoyancy flux, reveals that a baroclinic instability mechanism dominates over 5≲Q≲30, whereas stronger and weaker rotations are destabilised by vertical thermal gradients in the mean flow.

KW - Convection

KW - Instability

KW - Rotating flows

UR - http://www.scopus.com/inward/record.url?scp=84963700073&partnerID=8YFLogxK

U2 - 10.1017/jfm.2016.193

DO - 10.1017/jfm.2016.193

M3 - Article

VL - 795

SP - 1

EP - 35

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -