Brine
A certain person trying to make his way in the universe.
建國中學
暑假地球科學讀書會
大氣動力學
Presenter: 122527 Brine
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 \vec {g^*}
Centrifugal Force
\vec g = \vec{g^*} + \Omega^2 \vec R
Viscosity
x is parallel to the line of latitude
y is parallel 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}
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^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}]
= \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)
coriolis force
Using simple calculus
\delta P \approx P\sin\theta\omega\delta t
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
\frac{Dr}{Dt} = \omega \times r + \frac{dr}{dt}
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
coriolis force
Using simple calculus
coriolis force
Using simple calculus
\frac{Dr}{Dt} = \omega \times r + v
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
\frac{D^2r}{Dt^2} = \omega \times (\omega \times r + v) + \omega \times v + \frac{dv}{dt}
A = \frac{dv}{dt} + 2 \omega \times v + \omega^2r
coriolis force
Using simple calculus
\frac{Dr}{Dt} = \omega \times r + v
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
\frac{D^2r}{Dt^2} = \omega \times (\omega \times r + v) + \omega \times v + \frac{dv}{dt}
A = \frac{dv}{dt} + 2 \omega \times v + \omega^2r
coriolis force
Another approach - u
(\Omega + \frac{u}{R})^2 \vec{R}
= \Omega^2 \vec{R} + 2\Omega u \frac{\vec R}{R} + (\frac{u}{R})^2 \vec R
2 \Omega u \cos \phi
-2 \Omega u \sin \phi
coriolis force
Another approach - v
conservation\ of\ angular\ momentum
\Omega R^2 = (\Omega + \frac{\delta u}{R + \delta R})(R + \delta R)^2
\delta u \approx -2\Omega\delta R
\delta R = \delta y \sin \phi
\frac{du}{dt} = 2\Omega v \sin \phi
coriolis force
Another approach - w
conservation\ of\ angular\ momentum
\Omega R^2 = (\Omega + \frac{\delta u}{R + \delta R})(R + \delta R)^2
\delta u \approx -2\Omega\delta R
\delta R = \delta z \cos \phi
\frac{du}{dt} = -2\Omega w \cos \phi
coriolis force
Another approach - conclusion
\Biggl \{
\frac{d\vec V}{dt}
\Large\frac{du}{dt}\normalsize = 2\Omega v \sin \phi - 2 \Omega w \cos \phi
\Large\frac{dv}{dt}\normalsize = - 2 \Omega u \sin \phi
\Large\frac{dw}{dt}\normalsize = 2 \Omega u \cos \phi
coriolis force
Another approach - conclusion
\Biggl \{
\frac{d\vec V}{dt}
\Large\frac{du}{dt}\normalsize = 2\Omega v \sin \phi - 2 \Omega w \cos \phi
\Large\frac{dv}{dt}\normalsize = - 2 \Omega u \sin \phi
\Large\frac{dw}{dt}\normalsize = 2 \Omega u \cos \phi
f \equiv 2 \Omega\sin\phi
Equations
Introduction to some of the equations
Motion equatoins
\frac{d \vec V}{dt} = \vec g - \frac{1}{\rho} \nabla P + \nu \nabla^2 \vec V - 2 \vec \Omega \times \vec V
-2\vec \Omega \times \vec V = -2\Omega \begin{vmatrix}
\hat i & \hat j & \hat k\\
0 & \cos \phi & \sin \phi\\
u & v & w
\end{vmatrix}
spherical coordinate system
(x, y, z) \rightarrow (\theta( sometimes\ \lambda), \phi, r)
\Biggl \{
dx = r \cos \phi d\theta
dy = rd\phi
dz = dr
\vec V = (u, v, w) = \hat i u + \hat j v + \hat k w
spherical coordinate system
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
spherical coordinate system
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
\frac{d\hat i}{dt} = \frac{\partial \hat i}{\partial t} + u \frac{\partial \hat i}{\partial x}
\Large|\frac{\partial \hat i}{\partial x}| \normalsize= \displaystyle \lim_{\delta x \to 0} \frac{|\delta i|}{\delta x}
= \frac{|\hat i|\delta \theta}{r \cos \phi \delta \theta} = \frac{1}{r\cos \phi}
\delta \hat i = \sin \phi \hat j - \cos \phi \hat k
\delta \hat i
\sin \phi \hat j
- \cos \phi \hat k
spherical coordinate system
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
\frac{d\hat i}{dt} = \frac{\partial \hat i}{\partial t} + u \frac{\partial \hat i}{\partial x}
\Large|\frac{\partial \hat i}{\partial x}| \normalsize= \displaystyle \lim_{\delta x \to 0} \frac{|\delta i|}{\delta x}
= \frac{|\hat i|\delta \theta}{r \cos \phi \delta \theta} = \frac{1}{r\cos \phi}
u \frac{d\hat i}{dt} = u^2 \frac{\partial \hat i}{\partial x} = \frac{u^2}{r} \tan \phi \hat j - \frac{u^2}{r} \hat k
\frac{D\vec V_y}{Dt} = (\frac{Du}{Dt} + \frac{u^2}{r} \tan \phi + \frac{vw}{r}) \hat j
= - \frac{1}{\rho}\frac{\partial P}{\partial y} - 2\Omega u \sin\phi + F_y
\delta \hat i = \sin \phi \hat j - \cos \phi \hat k
\delta \hat i
\sin \phi \hat j
- \cos \phi \hat k
HORIZONTAL Approximation
geostrophic\ wind
-fv \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow v \approx \frac{1}{f\rho}\frac{\partial P}{\partial x}
fu \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow u \approx \frac{1}{f\rho}\frac{\partial P}{\partial y}
HORIZONTAL Approximation
geostrophic\ wind
-fv \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow v \approx \frac{1}{f\rho}\frac{\partial P}{\partial x}
fu \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow u \approx \frac{1}{f\rho}\frac{\partial P}{\partial y}
\frac{Du}{Dt} \approx - \frac{1}{\rho}\frac{\partial P}{\partial y} - \small{2\Omega u \sin\phi}
= - f(u - u_g)
= - \frac{1}{\rho}\frac{\partial P}{\partial y} - \small{2\Omega u \sin\phi} + F_y
\longrightarrow
(\frac{Du}{Dt} + \frac{u^2}{r} \small{\tan \phi} + \frac{vw}{r}) \hat j
Rossby number
- a tool to see whether the approximation is valid
Ro = \frac{a_{total}}{a_{coriolis}} = \frac{\frac{Du}{Dt}}{fu} = \frac{u / (l / u)}{fu} = \frac{u}{fl}
Ro < \frac{1}{10} \leftrightarrow valid
VERTICAL Approximation
\frac{dw}{dt} = - g - \frac{1}{\rho}\frac{\partial P}{\partial z}
\rightarrow
\frac{dw}{dt} \approx 0 \leftrightarrow equilibrium
tiny turbulence
g \approx - \frac{1}{\rho_0}\frac{\partial P_0}{\partial z}
P = P_0(z) + P'(x,y,z,t)
\rho = \rho_0(z) + \rho'(x,y,z,t)
\frac{dw}{dt} = - g - \frac{1}{\rho_0 + \rho'}\frac{\partial(P_0 + P')}{\partial z}
\forall \small x\ s.t.\ 0 < x << 1,\ \frac{1}{1 + x} \approx 1 - x
\frac{dw}{dt} \approx - \frac{1}{\rho_0}\frac{\partial P'}{\partial z} - \frac{\rho'}{\rho_0}g
continuity equation
x_{in} - x_{out} = - (\frac{\partial}{\partial x}\delta x) \rho u \delta y\delta z
...
- (\frac{\partial}{\partial x} \rho u + \frac{\partial}{\partial y} \rho v + \frac{\partial}{\partial z} \rho w)\delta x\delta y\delta z
= \frac{\partial \rho}{\partial t} \delta x\delta y\delta z
\frac{\partial \rho}{\partial t} +\small \nabla \cdot (\rho\vec V) = 0 \leftrightarrow \frac{D\rho}{Dt}+ \small \rho \nabla\cdot \vec V = 0
\nabla \cdot \vec v = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} +\frac{\partial v_z}{\partial z}...
another way
M = \rho V
\frac{DM}{Dt} = 0
\frac{1}{M}\frac{D}{Dt}\small(\rho V) = 0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \frac{1}{V}\frac{DV}{Dt}\small = 0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
(\frac{\partial }{\partial x} xyz = yz)\\
(\frac{\partial x}{\partial t}yz + \frac{\partial y}{\partial t}xz + \frac{\partial y}{\partial t}xz = \frac{DV}{Dt})
a bit of Manoeuvre
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
\rho = \rho_0(z) + \rho'(x,y,z,t)
\frac{1}{\rho_0+\rho'}\frac{D(\rho_0+\rho')}{Dt} + \nabla \cdot \vec V = 0
\large\frac{1}{\rho_0}\normalsize(\frac{\partial \rho'}{\partial t} + \small\vec V \cdot \nabla \rho') + \large\frac{1}{\rho_0}\frac{dz}{dt}\frac{d\rho_0}{dz} + \small\nabla \cdot \vec V \approx 0
\large\frac{1}{\rho_0}\normalsize w\frac{d\rho_0}{dz} + \small\nabla \cdot \vec V \approx 0
\frac{d\rho_0}{dz}w + \rho_0\small\nabla \cdot \vec V \approx 0
\frac{\partial\rho_0}{\partial x}u +\frac{\partial\rho_0}{\partial y}v +\frac{\partial \rho_0}{\partial z}w +
\rho_0 \frac{\partial u}{\partial x}+\rho_0 \frac{\partial v}{\partial y}+\rho_0 \frac{\partial w}{\partial z}+
\approx 0
\nabla \cdot (\rho_0 \vec V)
\approx 0
How similar
\nabla \cdot (\rho_0 \vec V)
\approx 0
incompressible\ flow
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
\delta \rho = 0
\small \nabla\cdot \vec V = 0
\small \nabla\cdot (\rho \vec V) = 0
\frac{d\rho_0}{dz}w + \rho_0\small\nabla \cdot \vec V \approx 0
Application
What can we do with all of that
Weather forecasting
of course- many models use the equations mentioned above
- prediction for weather / climate
what if
- predict what the world will look like with different conditions
thank you
I've got nothing left lmao
大氣動力學 Atmospheric Dynamics
By Brine
大氣動力學 Atmospheric Dynamics
The study of motion systems of meteorological importance, integrating observations at multiple locations and times and theories.
