建國中學

暑假地球科學讀書會

大氣動力學

Presenter: 12527 鄭竣陽

Index

Forces

  • pressure gradient force, PGF
  • gravitational force
  • viscosity
  • friction

apparent Forces

  • aka pseudo force
  • centrifugal force
  • coriolis force

whiteboard time

Drawing on computer is hard. Have mercy on me.

Pressure gradient force

  • assume there's an air parcel
a_x = -\frac{1}{\rho}\frac{\partial P}{\partial x\ }
a_y = -\frac{1}{\rho}\frac{\partial P}{\partial y\ }
a_z = -\frac{1}{\rho}\frac{\partial P}{\partial z\ }

Gravitational force

\vec{F_g} = \frac{GMm}{r^2}(\frac{\vec r}{r})
\frac{\vec{F_g}}{m} = \frac{GM}{r^2}(\frac{\vec r}{r}) \equiv g^*

Viscosity

x is parrallel to the line of latitude

y is parralel to the line of longitude

z is the altitude

\dot x \leftrightarrow u \\ \dot y \leftrightarrow v \\ \dot z \leftrightarrow w
F = \mu A \frac{\Delta u}{\Delta z}

(observed)

\tau_{z \to x} = \large \frac{F}{A} \normalsize = \mu \large \frac{\partial u}{\partial z}

Centrifugal Force

\vec g = \vec{g^*} + \Omega^2 \vec R

Viscosity

a_{z \to x} = \frac{(\tau_u - \tau_d)\Delta zA \\}{m}
\\ = \frac{\frac{\partial}{\partial z} \tau_{z \to x} \Delta x \Delta y \Delta z}{m}
\\ = \frac{1}{\rho} \frac{\partial}{\partial z} \tau_{z \to x}
\\ = \frac{\mu}{\rho} \frac{\partial}{\partial z} \frac{\partial u}{\partial z}
\\ = \nu \frac{\partial^2 u}{\partial z^2}
a_x = a_{x \to x} + a_{y \to x} + a_{z \to x}
= \nu [\frac{\partial^2 u}{\partial x} + \frac{\partial^2 u}{\partial y} + \frac{\partial^2 u}{\partial z}]
= \nu \nabla^2 u
\nabla = \displaystyle \sum_{i=1}^n \vec e_i {\partial \over \partial x_i} = \left({\partial \over \partial x_1}, \ldots, {\partial \over \partial x_n} \right)
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