Atmospheric Physics
Posted by Alex on June 1, 2012 • • Full article
Laws of Thermodynamics
1st Law of Thermo |
\[ \begin{eqnarray}
Q &=& \frac{dU}{dt} - W \\
J &=& c_v\frac{dT}{dt} + p\frac{d\alpha}{dt} \\
J &=& c_p\frac{dT}{dt} - \alpha\frac{dp}{dt}
\end{eqnarray} \]
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2nd Law of Thermo |
\[ \frac{dS}{dt} \geq \frac{Q}{T} \]
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3rd Law of Thermo |
\[ \lim_{T \to 0} S=0 \]
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Properties of Ideal Gases
Absolute Temperature |
\[ T = \frac{1}{3k_B} \left \langle mv^2 \right \rangle \]
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Absolute Pressure |
\[ p = \frac{1}{3} \frac{N}{V} \left \langle mv^2 \right \rangle \]
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Ideal Gas Law |
\[ pV = Nk_BT \]
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Dalton's Law |
\[ p = \sum_{i=1}^n p_i \]
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Exponential Decay |
\[ p \simeq p_0 \exp{ \left( -\frac{z}{H} \right) } \]
\[ \rho \simeq \rho_0 \exp{ \left( -\frac{z}{H} \right) } \]
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Scale Height |
\[ H = \frac{R_d \bar{T}}{g} \]
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Hypsometric Equation |
\[ z_2 - z_1 = \frac{R_d \bar{T}}{g} \ln{ \left( \frac{p_1}{p_2} \right) } \]
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Molecular Speeds
Maxwell-Boltzmann Function |
\[ f(v) = A \cdot 4\pi v^2 \exp{ \left( -\frac{mv^2}{2k_BT} \right) } \]
\[ A = \left( \frac{m}{2\pi k_BT} \right)^{3/2} \]
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Most Probable Speed |
\[ v_p = \sqrt{ \frac{2k_BT}{m} } \]
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Mean Speed |
\[ \left \langle v \right \rangle = \frac{2}{\sqrt{\pi}} v_p \]
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RMS Speed |
\[ v_{rms} = \sqrt{ \left \langle v^2 \right \rangle} = \sqrt{ \frac{3}{2} } v_p \]
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Escape Speed |
\[ v_{esc} = \sqrt{ 2gR_E } \]
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Intermolecular Distance |
\[ d = n^{-1/3} \]
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Material Properties
Specific Heat (Volume) |
\[ C_v = \left( \frac{dQ}{dT} \right)_{v} = \frac{dU}{dT} \]
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Specific Heat (Pressure) |
\[ C_p = \left( \frac{dQ}{dT} \right)_{p} = \frac{dH}{dT} \]
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Thermal Expansion |
\[ \alpha = \frac{1}{V}\frac{\partial V}{\partial T} \]
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Compressibility |
\[ \beta = -\frac{1}{V}\frac{\partial V}{\partial p} \]
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Thermodynamic Potentials
Gibbs Free Energy |
\[ G = H - TS \]
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Enthalpy |
\[ H = U + pV \]
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Helmholz Free Energy |
\[ F = U - TS \]
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Basic Relations
Thermodynamic Equilibrium |
Thermal Equilibrium |
\[ T' = T'' \] |
Mechanical Equilibrium |
\[ p' = p'' \] |
Chemical Equilibrium |
\[ \mu' = \mu'' \] |
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First Law of Thermo |
\[ \begin{eqnarray}
\delta Q &=& dU + \delta W \\
\delta q &=& c_vdT + pd\alpha \\
\delta q &=& c_pdT - \alpha dp \\
\end{eqnarray} \]
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Euler's Equation |
\[ TS = U + pV - \sum_{k=1}^{c} \mu_k \eta_k \]
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Gibbs-Duhem Relation |
\[ \sum \chi_k d\mu_k = \nu dp - sdT \]
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Gibbs Phase Rule |
\[ F = C - P + 2 \]
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Chemical Potential
(1 component) |
\[ d\mu = \nu dp - sdT \]
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Water Properties
Clausius-Clapyron Equation |
\[ \frac{dp}{dT} = \frac{L_v}{T\Delta \nu} \]
\[ e_s(T) = e_0 \exp{\left[ \frac{L_v}{R_v} \left( \frac{1}{T_0} - \frac{1}{T} \right) \right]} \]
\[ e_i(T) = e_0 \exp{\left[ \frac{L_s}{R_v} \left( \frac{1}{T_0} - \frac{1}{T} \right) \right]} \]
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Phase Lines |
\[ \frac{de_s}{dT} = \frac{L_v}{T\left( \nu_g - \nu_l \right)} > 0 \]
\[ \frac{de_i}{dT} = \frac{L_s}{T\left( \nu_g - \nu_s \right)} > 0 \]
\[ \frac{dp_{l/s}}{dT} = \frac{L_f}{T\left( \nu_l - \nu_s \right)} < 0 \]
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Dew Point |
\[ e = e_s(T_d) \]
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Frost Point |
\[ e = e_i(T_f) \]
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Bergeron-Findeison Process |
\[ e_s > e > e_i \]
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Evaporative Cooling |
\[ dq = -L_v dw \]
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Mixing Ratio |
\[ w = \frac{m_v}{m_d} = \frac{q}{1-q} \simeq \frac{\epsilon e}{p} \]
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Specific Humidity |
\[ q = \frac{m_v}{m_d+m_v} = \frac{w}{1+w} \]
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Virtual Temperature |
\[ T_v = T\left(1+0.61w\right) \]
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Potential Temperature |
\[ \theta = T \left( \frac{1000}{p} \right)^{R_d/c_p} \]
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Virtual Potential Temperature |
\[ \theta_v = T_v \left( \frac{1000}{p} \right)^{R_d/c_p} \]
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Equivalent Potential Temperature |
\[ \theta_e = \theta \left( \frac{L_v}{c_p} \frac{w_s}{T} \right) \]
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Lifted Condensation Level |
\[ z_{LCL} = \frac{T_0 - T_d}{\Gamma_d - \Gamma_{dew}} = \frac{T_0 - T_d}{8} \]
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Cloud Physics
Raoult's Law |
\[ e_{s,sol} = x_i e_{s,pure} \]
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Henry's Law |
\[ \frac{\left[ A\left(l\right) \right]}{\left[ A\left(g\right) \right]} = H_A^*RTL \]
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Mole Fraction |
\[ x_i = \frac{n_i}{n_{tot}} \]
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Kelvin Equation |
\[ e_{s,a} = a_w e_{s,\infty} \exp{\left( \frac{2\nu_w \sigma}{RTr} \right)} \]
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Köhler Equation |
\[ \frac{e'(r)}{e_s(T)} = 1 + \frac{a}{r}-\frac{b}{r^3} \]
\[ a = \frac{3.3\times10^{-5}}{T} \]
\[ b = \frac{4.3iM}{m_s} \]
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Stokes Flow |
\[ \left( \vec{u} \cdot \vec{\nabla} \right)\vec{u} = 0 \]
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Oseen Flow |
\[ \left( \vec{u} \cdot \vec{\nabla} \right)\vec{u} = \left( \vec{u_{\infty}} \cdot \vec{\nabla} \right)\vec{u} \]
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Growth by Diffusion |
\[ \frac{dM}{dt} = 4\pi a D_v \left( \rho_{v,\infty}-\rho_{v,a} \right) \]
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