The Brunt-Väisälä Frequency: a neat trick

Last updated 2018-01-04 11:07:53 SGT

(carrying on from the last time)

Consider the variation in the density as a function of position, parameterised by some quantity [\sigma]. We have that

[{\mathrm{d}\rho \over \mathrm{d} \sigma} = \left.\partial \rho \over \partial s\right|_X {\mathrm{d}s \over \mathrm{d} \sigma} + \left.\partial \rho \over \partial X\right|_s {\mathrm{d}X \over \mathrm{d} \sigma}, \tag{1}]

where [X] represents the set of configurational variables that affect the density (conventionally the temperature, pressure, and mean molecular weight; cf. our differential equation of state from the last time). This is clearly a separation of the derivative into adiabatic and nonadiabatic components.

This suggests that we consider the adiabatic evolution of a fluid packet as it is displaced (perhaps by some random fluctuation) from its equilibrium position, assuming one exists. Assuming that the packet is not too large, so that the gravitational field [\mathrm{g}] is approximately constant over its spatial extent, it feels an overall buoyant force given by [\mathbf{F} \sim \mathbf{g} V_\textrm{packet} \left(\rho_\textrm{packet} - \rho_0\right)]. By adiabaticity, for a sufficiently small displacement of the fluid packet from its equilibrium position we can write

[\mathbf{F} \sim \mathbf{g} V_\textrm{packet} \left(\left.\mathrm{d} \rho \over \mathrm{d} \sigma\right|_{\textrm{constant }S} - {\mathrm{d} \rho \over \mathrm{d} \sigma}\right) \delta \sigma.]

Since we are subtracting the adiabatic lapse from the total derivative of the density, by (1), what remains must then satisfy

[\mathbf{F} \sim -\mathbf{g} V_\textrm{packet} \left.\partial \rho \over \partial s\right|_X {\mathrm{d}s \over \mathrm{d}\sigma} \delta \sigma.]

In particular, for small displacements along the radial direction in a spherical configuration (so that [\mathbf{g} = -g \mathbf{e}_r]), we immediately obtain an expression for the Brunt-Väisälä frequency as

[N^2 = - {g \over \rho} \left.\partial \rho \over \partial s\right|_{P} {\mathrm{d} s \over \mathrm{d} r}.]

But we we found the last time that the gradient of the specific entropy yields the Ledoux criterion. Explicitly,

[\begin{aligned} N^2 &= {g \over \rho} \left.\partial \rho \over \partial s\right|_{P} \cdot {c_p \over H_p} \left(\nabla - \nabla_\mathrm{ad} - {\phi \over \delta}\nabla_\mu \right) \\ &= -{g \delta \over H_p} \left(\nabla - \nabla_\mathrm{ad} - {\phi \over \delta}\nabla_\mu \right),\end{aligned}]

which is the usual expression.

At this point I'm beginning to suspect that I've written too many blog posts complaining about how terrible the standard textbook pedagogy of this stuff is. I must say, however, that putting the adiabatic constraint front and centre makes the result much easier to digest conceptually. The comparison between a displaced fluid packet and its surroundings, which is an otherwise seemingly arbitrary decision, then emerges naturally, while at the same time serving purely as motivation, since no fiducial fluid packet need be subjected to analysis via differential quantities. It also follows immediately that the Brunt-Väisälä frequency vanishes when the density profile is entirely determined by the adiabatic lapse, which is nonobvious in the traditional approach.

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