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Hydrostatics

Hydrostatics

The focus of his course is fluid dynamics in which fluid parcels have interesting nonrivial motion. Nonethelss I would like to pause here and make a couple of statements about static fluid configurations in the presence of nontrivial forces.

1. Conservative force field

First, if the force per unit mass is conservative, then

\[ \frac{1}{\rho} {\vec F}_{ext} = {\vec \nabla} \Psi \]

Note this is not necessarily the same as \({\vec F}_{ext}\) being a gradient. Note also that it does include cases such as a constant vertical gravitational field for which \(\Psi = - g z\). In this case, if the fluid parcels are static,

\[ {\vec \nabla} p = \rho {\vec \nabla} \Psi \]

For constant \(\rho\) we would just have \(p = \rho \Psi\). More generally, taking the curl of both sides and recalling that the curl of the gradient vanishes,

\[ {\vec \nabla} \rho \times {\vec \nabla \Psi} = 0 \]

which is only possible if the gradients of \(\rho\) and \(\Psi\) are parallel, and thus their level lines coincide.

2. Example: Isothermal atmospere

Let us consider the case that the atmosphere is uniform in the directions horizontal to the ground, so that \(p,\rho\) depend on the vertical coordinate \(z\) only, and

\[ \partial_z p = - \rho g \]

If \(\rho\) is constant, then \(p(z) = p(0) - g \rho z\). If \(\rho\) is not constant but the atmosphere is an ideal gas at fixed temperature, \(p = \rho T/m\) is the ideal gas law, where \(m\) is the mass per molecule (for a monomolecular gas) so hat \(\rho/m\) is he number of molecules per unit volume. For constant \(T\) (“isothermal gas”), the hydrostatic equation becomes:

\[ \partial_z p = \frac{T}{m} \partial_z \rho = - \rho g \]

This has the exponential solution $\( \rho(z) = \rho(0) e^{-m g z/T} \Rightarrow p(z) = \frac{T \rho(0)}{m} e^{-mgz/T} \)\( where \)z = 0\( is the ground and \)z$ increases aboveground.

The actual atmosphere is more complex: the temperature itself drops in the troposhere up to the tropopause, around \(\sim 10\)km high, and then after remaining constant for another \(\sim 30\) km, begins to rise wih height in the stratosphere. The actual profile \(p(z)\) is somewhere between exponentially decaying and linearly decaying. In general, the atmosphere and ocean are stratified fluids with vertically varying densities.

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Vorticity