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
ax=−ρ1∂x ∂P
a_x = -\frac{1}{\rho}\frac{\partial P}{\partial x\ }
ay=−ρ1∂y ∂P
a_y = -\frac{1}{\rho}\frac{\partial P}{\partial y\ }
az=−ρ1∂z ∂P
a_z = -\frac{1}{\rho}\frac{\partial P}{\partial z\ }
Gravitational force
Fg=r2GMm(rr)
\vec{F_g} = \frac{GMm}{r^2}(\frac{\vec r}{r})
mFg=r2GM(rr)≡g∗
\frac{\vec{F_g}}{m} = \frac{GM}{r^2}(\frac{\vec r}{r}) \equiv \vec {g^*}
Centrifugal Force
g=g∗+Ω2R
\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
x˙↔uy˙↔vz˙↔w
\dot x \leftrightarrow u \\
\dot y \leftrightarrow v \\
\dot z \leftrightarrow w
F=μAΔzΔu
F = \mu A \frac{\Delta u}{\Delta z}
(observed)
τz→x=AF=μ∂z∂u
\tau_{z \to x} = \large \frac{F}{A} \normalsize = \mu \large \frac{\partial u}{\partial z}
Viscosity
az→x=m(τu−τd)ΔzA
a_{z \to x} = \frac{(\tau_u - \tau_d)\Delta zA \\}{m}
=m∂z∂τz→xΔxΔyΔz
\\ = \frac{\frac{\partial}{\partial z} \tau_{z \to x} \Delta x \Delta y \Delta z}{m}
=ρ1∂z∂τz→x
\\ = \frac{1}{\rho} \frac{\partial}{\partial z} \tau_{z \to x}
=ρμ∂z∂∂z∂u
\\ = \frac{\mu}{\rho} \frac{\partial}{\partial z} \frac{\partial u}{\partial z}
=ν∂z2∂2u
\\ = \nu \frac{\partial^2 u}{\partial z^2}
ax=ax→x+ay→x+az→x
a_x = a_{x \to x} + a_{y \to x} + a_{z \to x}
=ν[∂x2∂2u+∂y2∂2u+∂z2∂2u]
= \nu [\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}]
=ν∇2u
= \nu \nabla^2 u
∇=i=1∑nei∂xi∂=(∂x1∂,…,∂xn∂)
\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
δP≈Psinθωδt
\delta P \approx P\sin\theta\omega\delta t
DtDP=ω×P+dtdP
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
DtDr=ω×r+dtdr
\frac{Dr}{Dt} = \omega \times r + \frac{dr}{dt}
Dt2D2r=ω×DtDr+DtDv
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
coriolis force
Using simple calculus
coriolis force
Using simple calculus
DtDr=ω×r+v
\frac{Dr}{Dt} = \omega \times r + v
Dt2D2r=ω×DtDr+DtDv
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
DtDP=ω×P+dtdP
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
Dt2D2r=ω×(ω×r+v)+ω×v+dtdv
\frac{D^2r}{Dt^2} = \omega \times (\omega \times r + v) + \omega \times v + \frac{dv}{dt}
A=dtdv+2ω×v+ω2r
A = \frac{dv}{dt} + 2 \omega \times v + \omega^2r
coriolis force
Using simple calculus
DtDr=ω×r+v
\frac{Dr}{Dt} = \omega \times r + v
Dt2D2r=ω×DtDr+DtDv
\frac{D^2r}{Dt^2} = \omega \times \frac{Dr}{Dt} + \frac{Dv}{Dt}
DtDP=ω×P+dtdP
\frac{DP}{Dt} = \omega \times P + \frac{dP}{dt}
Dt2D2r=ω×(ω×r+v)+ω×v+dtdv
\frac{D^2r}{Dt^2} = \omega \times (\omega \times r + v) + \omega \times v + \frac{dv}{dt}
A=dtdv+2ω×v+ω2r
A = \frac{dv}{dt} + 2 \omega \times v + \omega^2r
coriolis force
Another approach - u
(Ω+Ru)2R
(\Omega + \frac{u}{R})^2 \vec{R}
=Ω2R+2ΩuRR+(Ru)2R
= \Omega^2 \vec{R} + 2\Omega u \frac{\vec R}{R} + (\frac{u}{R})^2 \vec R
2Ωucosϕ
2 \Omega u \cos \phi
−2Ωusinϕ
-2 \Omega u \sin \phi
coriolis force
Another approach - v
conservation of angular momentum
conservation\ of\ angular\ momentum
ΩR2=(Ω+R+δRδu)(R+δR)2
\Omega R^2 = (\Omega + \frac{\delta u}{R + \delta R})(R + \delta R)^2
δu≈−2ΩδR
\delta u \approx -2\Omega\delta R
δR=δysinϕ
\delta R = \delta y \sin \phi
dtdu=2Ωvsinϕ
\frac{du}{dt} = 2\Omega v \sin \phi
coriolis force
Another approach - w
conservation of angular momentum
conservation\ of\ angular\ momentum
ΩR2=(Ω+R+δRδu)(R+δR)2
\Omega R^2 = (\Omega + \frac{\delta u}{R + \delta R})(R + \delta R)^2
δu≈−2ΩδR
\delta u \approx -2\Omega\delta R
δR=δzcosϕ
\delta R = \delta z \cos \phi
dtdu=−2Ωwcosϕ
\frac{du}{dt} = -2\Omega w \cos \phi
coriolis force
Another approach - conclusion
{
\Biggl \{
dtdV
\frac{d\vec V}{dt}
dtdu=2Ωvsinϕ−2Ωwcosϕ
\Large\frac{du}{dt}\normalsize = 2\Omega v \sin \phi - 2 \Omega w \cos \phi
dtdv=−2Ωusinϕ
\Large\frac{dv}{dt}\normalsize = - 2 \Omega u \sin \phi
dtdw=2Ωucosϕ
\Large\frac{dw}{dt}\normalsize = 2 \Omega u \cos \phi
coriolis force
Another approach - conclusion
{
\Biggl \{
dtdV
\frac{d\vec V}{dt}
dtdu=2Ωvsinϕ−2Ωwcosϕ
\Large\frac{du}{dt}\normalsize = 2\Omega v \sin \phi - 2 \Omega w \cos \phi
dtdv=−2Ωusinϕ
\Large\frac{dv}{dt}\normalsize = - 2 \Omega u \sin \phi
dtdw=2Ωucosϕ
\Large\frac{dw}{dt}\normalsize = 2 \Omega u \cos \phi
f≡2Ωsinϕ
f \equiv 2 \Omega\sin\phi
Equations
Introduction to some of the equations
Motion equatoins
dtdV=g−ρ1∇P+ν∇2V−2Ω×V
\frac{d \vec V}{dt} = \vec g - \frac{1}{\rho} \nabla P + \nu \nabla^2 \vec V - 2 \vec \Omega \times \vec V
−2Ω×V=−2Ωi^0uj^cosϕvk^sinϕw
-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)→(θ(sometimes λ),ϕ,r)
(x, y, z) \rightarrow (\theta( sometimes\ \lambda), \phi, r)
{
\Biggl \{
dx=rcosϕdθ
dx = r \cos \phi d\theta
dy=rdϕ
dy = rd\phi
dz=dr
dz = dr
V=(u,v,w)=i^u+j^v+k^w
\vec V = (u, v, w) = \hat i u + \hat j v + \hat k w
spherical coordinate system
dtd(i^u)=i^dtdu+udtdi^
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
spherical coordinate system
dtd(i^u)=i^dtdu+udtdi^
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
dtdi^=∂t∂i^+u∂x∂i^
\frac{d\hat i}{dt} = \frac{\partial \hat i}{\partial t} + u \frac{\partial \hat i}{\partial x}
∣∂x∂i^∣=δx→0limδx∣δi∣=rcosϕδθ∣i^∣δθ=rcosϕ1
\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}
δi^=sinϕj^−cosϕk^
\delta \hat i = \sin \phi \hat j - \cos \phi \hat k
δi^
\delta \hat i
sinϕj^
\sin \phi \hat j
−cosϕk^
- \cos \phi \hat k
spherical coordinate system
dtd(i^u)=i^dtdu+udtdi^
\frac{d(\hat i u)}{dt} = \hat i \frac{du}{dt} + u \frac{d\hat i}{dt}
dtdi^=∂t∂i^+u∂x∂i^
\frac{d\hat i}{dt} = \frac{\partial \hat i}{\partial t} + u \frac{\partial \hat i}{\partial x}
∣∂x∂i^∣=δx→0limδx∣δi∣=rcosϕδθ∣i^∣δθ=rcosϕ1
\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}
udtdi^=u2∂x∂i^=ru2tanϕj^−ru2k^
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
DtDVy=(DtDu+ru2tanϕ+rvw)j^
\frac{D\vec V_y}{Dt} = (\frac{Du}{Dt} + \frac{u^2}{r} \tan \phi + \frac{vw}{r}) \hat j
=−ρ1∂y∂P−2Ωusinϕ+Fy
= - \frac{1}{\rho}\frac{\partial P}{\partial y} - 2\Omega u \sin\phi + F_y
δi^=sinϕj^−cosϕk^
\delta \hat i = \sin \phi \hat j - \cos \phi \hat k
δi^
\delta \hat i
sinϕj^
\sin \phi \hat j
−cosϕk^
- \cos \phi \hat k
HORIZONTAL Approximation
geostrophic wind
geostrophic\ wind
−fv≈−ρ1∂x∂P→v≈fρ1∂x∂P
-fv \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow v \approx \frac{1}{f\rho}\frac{\partial P}{\partial x}
fu≈−ρ1∂x∂P→u≈fρ1∂y∂P
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
geostrophic\ wind
−fv≈−ρ1∂x∂P→v≈fρ1∂x∂P
-fv \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow v \approx \frac{1}{f\rho}\frac{\partial P}{\partial x}
fu≈−ρ1∂x∂P→u≈fρ1∂y∂P
fu \approx -\frac{1}{\rho}\frac{\partial P}{\partial x} \rightarrow u \approx \frac{1}{f\rho}\frac{\partial P}{\partial y}
DtDu≈−ρ1∂y∂P−2Ωusinϕ=−f(u−ug)
\frac{Du}{Dt} \approx - \frac{1}{\rho}\frac{\partial P}{\partial y} - \small{2\Omega u \sin\phi}
= - f(u - u_g)
=−ρ1∂y∂P−2Ωusinϕ+Fy
= - \frac{1}{\rho}\frac{\partial P}{\partial y} - \small{2\Omega u \sin\phi} + F_y
⟶
\longrightarrow
(DtDu+ru2tanϕ+rvw)j^
(\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=acoriolisatotal=fuDtDu=fuu/(l/u)=flu
Ro = \frac{a_{total}}{a_{coriolis}} = \frac{\frac{Du}{Dt}}{fu} = \frac{u / (l / u)}{fu} = \frac{u}{fl}
Ro<101↔valid
Ro < \frac{1}{10} \leftrightarrow valid
VERTICAL Approximation
dtdw=−g−ρ1∂z∂P
\frac{dw}{dt} = - g - \frac{1}{\rho}\frac{\partial P}{\partial z}
→
\rightarrow
dtdw≈0↔equilibrium
\frac{dw}{dt} \approx 0 \leftrightarrow equilibrium
tiny turbulence
g≈−ρ01∂z∂P0
g \approx - \frac{1}{\rho_0}\frac{\partial P_0}{\partial z}
P=P0(z)+P′(x,y,z,t)
P = P_0(z) + P'(x,y,z,t)
ρ=ρ0(z)+ρ′(x,y,z,t)
\rho = \rho_0(z) + \rho'(x,y,z,t)
dtdw=−g−ρ0+ρ′1∂z∂(P0+P′)
\frac{dw}{dt} = - g - \frac{1}{\rho_0 + \rho'}\frac{\partial(P_0 + P')}{\partial z}
∀x s.t. 0<x<<1, 1+x1≈1−x
\forall \small x\ s.t.\ 0 < x << 1,\ \frac{1}{1 + x} \approx 1 - x
dtdw≈−ρ01∂z∂P′−ρ0ρ′g
\frac{dw}{dt} \approx - \frac{1}{\rho_0}\frac{\partial P'}{\partial z} - \frac{\rho'}{\rho_0}g
continuity equation
xin−xout=−(∂x∂δx)ρuδyδz
x_{in} - x_{out} = - (\frac{\partial}{\partial x}\delta x) \rho u \delta y\delta z
...
...
−(∂x∂ρu+∂y∂ρv+∂z∂ρw)δxδyδz=∂t∂ρδxδyδz
- (\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
∂t∂ρ+∇⋅(ρV)=0↔DtDρ+ρ∇⋅V=0
\frac{\partial \rho}{\partial t} +\small \nabla \cdot (\rho\vec V) = 0 \leftrightarrow \frac{D\rho}{Dt}+ \small \rho \nabla\cdot \vec V = 0
∇⋅v=∂x∂vx+∂y∂vy+∂z∂vz...
\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=ρV
M = \rho V
DtDM=0
\frac{DM}{Dt} = 0
M1DtD(ρV)=0
\frac{1}{M}\frac{D}{Dt}\small(\rho V) = 0
ρ1DtDρ+V1DtDV=0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \frac{1}{V}\frac{DV}{Dt}\small = 0
ρ1DtDρ+∇⋅V=0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
(∂x∂xyz=yz)(∂t∂xyz+∂t∂yxz+∂t∂yxz=DtDV)
(\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
ρ1DtDρ+∇⋅V=0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
ρ=ρ0(z)+ρ′(x,y,z,t)
\rho = \rho_0(z) + \rho'(x,y,z,t)
ρ0+ρ′1DtD(ρ0+ρ′)+∇⋅V=0
\frac{1}{\rho_0+\rho'}\frac{D(\rho_0+\rho')}{Dt} + \nabla \cdot \vec V = 0
ρ01(∂t∂ρ′+V⋅∇ρ′)+ρ01dtdzdzdρ0+∇⋅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
ρ01wdzdρ0+∇⋅V≈0
\large\frac{1}{\rho_0}\normalsize w\frac{d\rho_0}{dz} + \small\nabla \cdot \vec V \approx 0
dzdρ0w+ρ0∇⋅V≈0
\frac{d\rho_0}{dz}w + \rho_0\small\nabla \cdot \vec V \approx 0
∂x∂ρ0u+∂y∂ρ0v+∂z∂ρ0w+ρ0∂x∂u+ρ0∂y∂v+ρ0∂z∂w+≈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
∇⋅(ρ0V)≈0
\nabla \cdot (\rho_0 \vec V)
\approx 0
How similar
∇⋅(ρ0V)≈0
\nabla \cdot (\rho_0 \vec V)
\approx 0
incompressible flow
incompressible\ flow
ρ1DtDρ+∇⋅V=0
\frac{1}{\rho}\frac{D\rho}{Dt}+ \small \nabla\cdot \vec V = 0
δρ=0
\delta \rho = 0
∇⋅V=0
\small \nabla\cdot \vec V = 0
∇⋅(ρV)=0
\small \nabla\cdot (\rho \vec V) = 0
dzdρ0w+ρ0∇⋅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
建國中學 暑假地球科學讀書會 大氣動力學 Presenter: 1 22527 Brine
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.
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