Between two free boundaries a distance apart

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Earth, Climate & Energy Physics · Seismology, tectonics and geodynamics

The Solid Earth, Ch Plates, creep and convection: the mantle in motion

Problem

Between two free boundaries a distance

d

apart, eliminating the pressure and velocity from the slow-flow and heat equations gives one equation for the dimensionless temperature disturbance

\theta

, with

z' = z/d \in [0,1]

and

K = kd

the dimensionless horizontal wavenumber of a convection roll:

\left(\frac{d^2}{dz'^2} - K^2\right)^3 \theta = -\mathrm{Ra}\,K^2\theta,

with

\theta = d^2\theta/dz'^2 = d^4\theta/dz'^4 = 0

z' = 0

and

z' = 1

(free boundaries), where

\mathrm{Ra}

is the Rayleigh number of the layer. Trying

\theta = \theta_0\sin(\pi z')\cos(kx)

gives the neutral curve

\mathrm{Ra}(K) = (\pi^2 + K^2)^3/K^2

, minimised at

\mathrm{Ra}_c = 27\pi^4/4 = 657.5

when

K = \pi/\sqrt2

. ((a)) Repeat the calculation with the trial function

\theta = \theta_0\sin(2\pi z')\cos(kx)

, a mode with one extra internal node. Derive the neutral curve

\mathrm{Ra}(K)

for this mode and minimise it over

K

to find its critical value and the wavenumber at which it occurs. ((b)) State the ratio of the critical value found in (a) to

657.5

, and explain physically, in terms of the vertical structure each mode describes, why the

n = 1

mode

\sin(\pi z')

, and not this one, sets the true threshold of instability.

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