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The linearized equations

Now we put in the Boussinesq equations the following perturbed quantities:

$\displaystyle \boldsymbol {u}=\epsilon \boldsymbol {u'}(x,y,z,t),\;T=T_0(z)+\epsilon \theta'(x,y,z,t)$ and $\displaystyle p=P_0(z)+\epsilon p'(x,y,z,t)

Collecting the coefficient of $ \epsilon ^0$ we get the basic equlibrium state. Collecting the coefficeint of $ \epsilon ^1$ we arrived at the following linearized equations.

$\displaystyle \nabla\cdot \boldsymbol {u'}=0$ (7)

$\displaystyle \frac{\ensuremath{\partial}\boldsymbol {u'}}{\ensuremath{\partial...
...rac{1}{\rho_b}\nabla p' + \alpha g \theta' \hat{k} + \nu \Delta\boldsymbol {u'}$ (8)

$\displaystyle \frac{\ensuremath{\partial}\theta'}{\ensuremath{\partial}t} - \beta w' = \kappa \Delta\theta'$ (9)

A/C for Homepage of Dr. S Ghorai 2003-01-16