Problem 3: Convection with modified vertical acceleration

We consider the Boussinesq linear system with vertical momentum modified by the parameter $\alpha$:

$$\alpha^2 \frac{D w}{D t} = -\frac{\partial \phi}{\partial z} + b,$$ (25)

while the remaining linearized equations (about rest, constant stratification) are the usual ones:

$$\frac{\partial u}{\partial t} = -\frac{\partial \phi}{\partial x}, \qquad ( )$$ (26)

$$\alpha^2\frac{\partial w}{\partial t} = -\frac{\partial \phi}{\partial z} + b, \qquad ( )$$ (27)

$$\frac{\partial b}{\partial t} + N^2 w = 0, \qquad ( )$$ (28)

$$\frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} = 0. \qquad ( )$$ (29)

Seeking plane-wave solutions $\propto e^{i(kx + mz - \omega t)}$, substitute derivatives: $\partial_t\mapsto -i\omega$, $\partial_x\mapsto i k$, $\partial_z\mapsto i m$.

From horizontal momentum:

$$- i\omega u = - i k \phi \;\Rightarrow\; \phi = \frac{\omega}{k} u.$$ (30)

From incompressibility: $k u + m w = 0\;\Rightarrow\; u = -\frac{m}{k} w$. Substitute into ():

$$\phi = -\frac{\omega m}{k^2} w.$$ (31)

From buoyancy: $-i\omega b + N^2 w = 0 \Rightarrow b = \dfrac{N^2}{i\omega}w = -i\dfrac{N^2}{\omega} w$. From vertical momentum: $-i\omega\alpha^2 w = - i m \phi + b$. Substitute for $\phi$ and $b$ using ():

$$- i\omega\alpha^2 w = - i m\left(-\frac{\omega m}{k^2} w\right) - i\frac{N^2}{\omega} w$$ (32)

$$= i \frac{\omega m^2}{k^2} w - i\frac{N^2}{\omega} w.$$ (33)

Multiply through by $-i$ and divide by $w$ (nonzero for waves) to obtain

$$\omega \alpha^2 = -\frac{\omega m^2}{k^2} + \frac{N^2}{\omega}.$$ (34)

Multiply by $\omega$ and rearrange:

$$\omega^2\left(\alpha^2 + \frac{m^2}{k^2}\right) = N^2.$$ (35)

Thus the dispersion relation is

$$\boxed{\;\omega^2 = \dfrac{N^2 k^2}{\alpha^2 k^2 + m^2}\; }.$$ (36)

This is the general dispersion relation for the modified vertical inertia parameter $\alpha$. Special cases:

• Hydrostatic limit $\alpha=0$: $$\omega^2 = \frac{N^2 k^2}{m^2} \; (\Longrightarrow \omega = \pm N\,\frac{k}{m})$$, the hydrostatic gravity-wave relation.
• Full nonhydrostatic Boussinesq $\alpha=1$: $$\omega^2 = \frac{N^2 k^2}{k^2 + m^2}$$, the standard dispersion relation for internal gravity waves (bounded by $\vert\omega\vert<N$).