Atmospheric Dynamics
Mathematical Relations
Taylor Series | \[ f(x) = f(a) + \frac{\partial f}{\partial x} \bigg|_{a}^{} \frac{(x-a)}{1!} + \frac{\partial^2 f}{\partial x^2} \bigg|_{a}^{} \frac{(x-a)^2}{2!} + \cdots \] |
Geometric Power Series | \[ \frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots \] |
Gauss' Theorem | \[ \iint_{S}^{} \vec{F} \cdot \vec{n}\ dS = \iiint_{V}^{} (\vec{\nabla} \cdot \vec{F})\ dV \] \[ Divergence \approx \frac{Flux}{Volume} \] |
Stokes' Theorem | \[ \oint_{\Gamma}^{} \vec{U} \cdot d\vec{R} = \iint_{S}^{} (\vec{\nabla} \times \vec{U})\cdot \vec{n}\ dS \] \[ Vorticity \approx \frac{Circulation}{Area} \] |
Dot Product | \[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 \] \[ \vec{a} \cdot \vec{b} = \left| \vec{a} \right| \left| \vec{b} \right| \cos{\theta} \] |
Cross Product | \[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\ \] \[ \vec{a} \times \vec{b} = \left \langle a_2 b_3 - a_3 b_2,\ a_3 b_1 - a_1 b_3,\ a_1 b_2 - a_2 b_1 \right \rangle \] \[ \left| \vec{a} \times \vec{b} \right| = \left| \vec{a} \right | \left | \vec{b} \right| \sin{\theta} \] |
Material Derivative | \[ \frac{d\vec{U}}{dt} = \frac{\partial \vec{U}}{\partial t} + u\frac{\partial \vec{U}}{\partial x} + v\frac{\partial \vec{U}}{\partial y} + w\frac{\partial \vec{U}}{\partial z} = \frac{\partial \vec{U}}{\partial t} + \left( \vec{U} \cdot \vec{\nabla} \right) \vec{U} \] |
Jacobian | \[ J(A,B) = \partial_x A \partial_y B - \partial_y A \partial_x B \] |
Laplacian | \[ \nabla^2 \psi = \vec{\nabla} \cdot (\nabla \psi) = \partial_{xx}\psi + \partial_{yy}\psi \] |
Euler's Formula | \[ e^{ i\theta } = \cos{\theta} + i\sin{\theta} \] |
Einstein Notation | \[ \epsilon_{ij3} = \epsilon_{123} + \epsilon_{213} = (1) + (-1) \] |
Complex Conjugate | \[ \left( a+ib \right) \left( a-ib \right) = a^2+b^2 \] |
Wave Equation | \[ \frac{\partial^2 \psi}{\partial x^2} = c^2 \left( \nabla^2 \psi \right) \] |
Wave Solution | \[ \psi' = Re \left \{ \psi_0 e^{i\left(kx+ly+mz-\omega t\right)} \right \} \] |
Heat Diffusion Equation | \[ \frac{\partial T}{\partial t} = \vec{\kappa} \nabla^2 T = k_1 \partial_{xx}T + k_2 \partial_{yy}T + k_3 \partial_{zz}T \] |
Second Order DEQ | \[ ay'' + by' + cy = 0\ \ \ (Differential\ Equation) \] \[ ar^2 + br + c = 0\ \ \ (Characteristic\ Equation) \] |
Quadratic Formula | \[ r = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a} \] |
DEQ General Solutions | \[ y = C_1e^{rx} + C_2e^{-rx}\ \ \ (Different\ Roots) \] \[ y = C_1e^{rx} + C_2xe^{-rx}\ \ \ (Same\ Roots) \] \[ y = \left[ C_1 \cos{bx} + C_2 \sin{bx} \right] e^{ax}\ \ \ (Complex\ Roots) \] |
Equations of Motion
Navier-Stokes (Differential) | \[ \partial_tu + u\partial_xu + v\partial_yu + w\partial_zu = -\frac{1}{\rho}\frac{\partial p}{\partial x} + fv + \nu\nabla^2 u \] \[ \partial_tv + u\partial_xv + v\partial_yv + w\partial_zv = -\frac{1}{\rho}\frac{\partial p}{\partial y} - fu + \nu\nabla^2 v \] \[ 0 = -\frac{1}{\rho}\frac{\partial p}{\partial z} -g \] |
Navier-Stokes (Vector) | \[ \frac{\partial \vec{U}}{\partial t} + (\vec{U} \cdot \vec{\nabla})\vec{U} = -\frac{1}{\rho}\vec{\nabla}p -\vec{\nabla}\Phi - 2(\vec{\Omega} \times \vec{U}) + \nu\nabla^2 \vec{U} \] \[ Local + Advective = PGF + Gravity + Coriolis + Viscosity \] |
Effects of Rotation
Non-Inertial Acceleration | \[ \left( \frac{d \vec{U}}{dt} \right)_R = \left( \frac{d \vec{U}}{dt} \right)_I - 2(\vec{\Omega} \times \vec{U}) - \vec{\Omega} \times (\vec{\Omega} \times \vec{R}) \] |
Absolute Velocity | \[ \vec{U_a} = \vec{U} + \vec{\Omega} \times \vec{R} \] |
Relative Vorticity | \[ \zeta = \left[ \vec{\nabla} \times \vec{U} \right] \cdot \hat{k} = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \] |
Planetary Vorticity | \[ f = \left[ \vec{\nabla} \times (\vec{\Omega} \times \vec{R}) \right] \cdot \hat{k} = 2\vec{\Omega} \cdot \hat{k} = 2\Omega \sin{\phi} \] |
Absolute Vorticity | \[ \zeta_a = \zeta + f \] |
Potential Vorticity | \[ PV = \frac{ \zeta + f}{H} \] |
Ertel's Potential Vorticity | \[ EPV = \frac{ \vec{\zeta_a} \cdot \vec{\nabla} \theta}{\rho} \] |
Ertel's Theorem | \[ \frac{d}{dt} \left( \frac{ \vec{\zeta_a} \cdot \vec{\nabla} \theta}{\rho} \right) = \frac{d}{dt} \left( \frac{ \vec{\zeta_a} \cdot \vec{n}}{\rho \ \delta l} \right) = 0 \] |
Thermodynamics
First Law of Thermo | \[ \begin{eqnarray} Q &=& \frac{dU}{dt} - W \\ J &=& c_v\frac{dT}{dt} + p\frac{d\alpha}{dt} \\ J &=& c_p\frac{dT}{dt} - \alpha\frac{dp}{dt} \end{eqnarray} \] |
Second Law of Thermo | \[ \frac{dS}{dt} \geq \frac{Q}{T} \] |
Third Law of Thermo | \[ \lim_{T \to 0} S=0 \] |
Absolute Temperature | \[ T = \frac{1}{3k_B} \left \langle mv^2 \right \rangle \] |
Absolute Pressure | \[ p = \frac{1}{3} \frac{N}{V} \left \langle mv^2 \right \rangle \] |
Ideal Gas Law | \[ pV = Nk_BT \] |
Scale Height | \[ H = \frac{R_d \bar{T}}{g} \] |
Exponential Decay | \[ p \simeq p_0 \exp{ \left( -\frac{z}{H} \right) } \] \[ \rho \simeq \rho_0 \exp{ \left( -\frac{z}{H} \right) } \] |
Hypsometric Equation | \[ z_2 - z_1 = \frac{R_d \bar{T}}{g} \ln{ \left( \frac{p_1}{p_2} \right) } \] |
Maxwell-Boltzmann Function | \[ f(v) = A \cdot 4\pi v^2 \exp{ \left( -\frac{mv^2}{2k_BT} \right) } \] \[ A = \left( \frac{m}{2\pi k_BT} \right)^{3/2} \] | Most Probable Speed | \[ v_p = \sqrt{ \frac{2k_BT}{m} } \] |
Mean Speed | \[ \left \langle v \right \rangle = \frac{2}{\sqrt{\pi}} v_p \] |
RMS Speed | \[ v_{rms} = \sqrt{ \left \langle v^2 \right \rangle} = \sqrt{ \frac{3}{2} } v_p \] |
Escape Speed | \[ v_{esc} = \sqrt{ 2gR_E } \] |
Specific Heat (Volume) | \[ C_v = \left( \frac{dQ}{dT} \right)_{v} = \frac{dU}{dT} \] |
Specific Heat (Pressure) | \[ C_p = \left( \frac{dQ}{dT} \right)_{p} = \frac{dH}{dT} \] |
Enthalpy | \[ H = U + pV \] |
Gibbs Free Energy | \[ G = H - TS \] |
Poisson's Relation | \[ \theta = T \left( \frac{1000}{p} \right)^{R_d/c_p} \] |
Flow Properties
Scale Analysis |
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Froude Number | \[ Fr = \frac{\lvert \frac{U^2}{H} \rvert}{\lvert g \rvert} = \frac{U}{\sqrt{gH}} = \left( \frac{L}{L_D} \right)^2 \] | |||
Richardson Number | \[ Ri = \frac{N^2}{\left( \frac{du}{dz} \right)^2} = \frac{gH}{U^2} = \left[ \sqrt{Fr} \right]^{-1} \] | |||
Reynolds Number | \[ Re = \frac{\lvert u\frac{\partial u}{\partial x} \rvert}{\lvert \nu\nabla^2 u \rvert} = \frac{UL}{\nu} \] | |||
Rossby Number |
\[ Ro = \frac{\lvert u\frac{\partial u}{\partial x} \rvert}{\lvert fu \rvert} = \frac{U}{fL} \]
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Temporal Rossby Number | \[ Ro_{T} = \frac{U}{fT} \] | |||
Burger Number | \[ Bu = \left( \frac{Ro}{Fr} \right)^2 \] | |||
Ekman Number | \[ Ek = \frac{\nu_{E}}{fH^2} \] | |||
Geostropic Balance | \[ Ro = \frac{U}{fL} \ll 1 \] | |||
Bowen Ratio | \[B_e = \frac{SH}{LH} = \frac{c_p}{L_v} \left( \frac{\partial q}{\partial T} \right)^{-1} \] | |||
Aspect Ratio | \[ \delta = \frac{H}{L} \] | |||
Hydrostatic Balance | \[ \delta = \frac{H}{L} \ll 1 \] | |||
Flow Decomposition | \[ \vec{U} = \vec{U_D} + \vec{U_E} + \vec{U_R} \] | |||
Deformation Flow | \[ \vec{U_D} = \frac{1}{2} \vec{D} \cdot \vec{x} = \frac{1}{2} \begin{vmatrix} T & H \\ H & -T \end{vmatrix}\ \vec{x} \] | |||
Expansion Flow | \[ \vec{U_E} = \frac{1}{2} \vec{E} \cdot \vec{x} = \frac{1}{2} \begin{vmatrix} \vec{\nabla} \cdot \vec{U} & 0 \\ 0 & \vec{\nabla} \cdot \vec{U} \end{vmatrix}\ \vec{x} \] | |||
Rotational Flow | \[ \vec{U_R} = \frac{1}{2} \vec{R} \cdot \vec{x} = \frac{1}{2} \begin{vmatrix} \zeta & 0 \\ 0 & -\zeta \end{vmatrix}\ \vec{x} \] | |||
Stretching Deformation | \[ T = \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \] | |||
Shearing Deformation | \[ H = \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \] | |||
Basic Dynamics
Divergence | \[ \vec{\nabla} \cdot \vec{U} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \] | |||
Advection | \[ -\vec{U} \cdot \nabla T = -u\frac{\partial T}{\partial x} - v\frac{\partial T}{\partial y} \] | |||
Dry Adiabatic Lapse Rate | \[ \Gamma_d = \frac{g}{c_p} \] | |||
Coriolis Force | \[ -2(\vec{\Omega} \times \vec{U}) = -2\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & \Omega\cos\phi & \Omega\sin\phi \\ u & v & w \end{vmatrix} \simeq \left \langle -fv,\ fu,\ 0 \right \rangle \] | |||
Centrifugal Force | \[ -\vec{\Omega} \times (\vec{\Omega} \times \vec{R}) = \Omega^2 \vec{R} \] | |||
Effective Gravity | \[ \vec{g} = \vec{g^{*}} + \Omega^2 \vec{R} = \left \langle 0,\ 0,-g \right \rangle \] | |||
Coriolis Parameter | \[ f = 2\Omega \sin \phi \] | |||
Geopotential Height | \[ \Phi = \left \langle 0,\ 0,\ gz \right \rangle \] \[ \frac{\partial \Phi}{\partial p} = -\alpha = -\frac{R_dT}{p} \] | |||
Hypsometric Equation | \[ z_2 - z_1 = \frac{1}{g} \int_{p_1}^{p_2} \frac{\partial \Phi}{\partial p} dp = \frac{R_d \bar{T}}{g} \ln{\left (\frac{p_1}{p_2}\right )} \] | |||
Geostrophic Wind | \[ u_g = -\frac{1}{f\rho} \frac{\partial p}{\partial y} = -\frac{g}{f_0} \frac{\partial \eta}{\partial y} = -\left( \frac{\partial \Phi}{\partial y} \right)_p \] \[ v_g = \frac{1}{f\rho} \frac{\partial p}{\partial x} = \frac{g}{f_0} \frac{\partial \eta}{\partial x} = \left( \frac{\partial \Phi}{\partial x} \right)_p \] | |||
Thermal Wind | \[ \frac{\partial u_g}{\partial z} = -\frac{g}{fT} \left( \frac{\partial T}{\partial y} \right)_p \] \[ \frac{\partial v_g}{\partial z} = \frac{g}{fT} \left( \frac{\partial T}{\partial x} \right)_p \] | |||
Brunt-Väisälä Frequency | \[ N^2 = -\frac{g}{\theta} \frac{\partial \theta}{\partial z} \] | |||
Balanced Flow |
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Mass Continuity | \[ \frac{d \rho}{dt} + \rho \vec{\nabla} \cdot \vec{U} = 0 \] | |||
Boussinesq Approximation | \[ \frac{d\rho}{dt} = 0 \] \[ \rho = \rho_0 \] | |||
Primative Equations | \[ \partial_tu + u\partial_xu + v\partial_yu -\omega\partial_pu - \partial_x\Phi - fv = F_x \] \[ \partial_tv + u\partial_xv + v\partial_yv -\omega\partial_pv - \partial_y\Phi + fu = F_y \] \[ \partial_p\Phi = -\frac{RT}{p} \] \[ \partial_tT + u\partial_xT + v\partial_yT + \omega \left(\frac{\kappa T}{p} - \frac{\partial T}{\partial p} \right) + \frac{\dot{H}}{c_p} \] \[ \partial_p\omega = - \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) \] | |||
Phase Speed | \[ \vec{c} = \left \langle \frac{\omega}{k},\ \frac{\omega}{l},\ \frac{\omega}{m} \right \rangle \] | |||
Group Velocity | \[ \vec{c_g} = \left \langle \frac{\partial \omega}{\partial k},\ \frac{\partial \omega}{\partial l},\ \frac{\partial \omega}{\partial m} \right \rangle \] | |||
Wavenumber | \[ \vec{\kappa} = \left \langle k,\ l,\ m \right \rangle = \left \langle \frac{2\pi}{\lambda_x},\ \frac{2\pi}{\lambda_y},\ \frac{2\pi}{\lambda_z} \right \rangle\] \[ \kappa^2 = k^2 + l^2 + m^2 \] | |||
Wave Dispersion | \[ \omega = \omega (k)\ \ \ (Dispersive) \] \[ \omega \neq \omega (k)\ \ \ (Non-Dispersive) \] | |||
Symmetric Instability | \[ \left( \frac{\delta z}{\delta y} \right)_\theta > \left( \frac{\delta z}{\delta y} \right)_M \] \[ M = U - fy \] | |||
Shallow Water Equations
Plane Approximations |
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Beta | \[ \beta = \frac{df}{dy} = \frac{2\Omega}{a}\cos\theta_0 \] | |||
1-Layer SWE | \[ \partial_tu + u\partial_xu + v\partial_yu - fv = -g\partial_x\eta \] \[ \partial_tv + u\partial_xv + v\partial_yv + fu = -g\partial_y\eta \] \[ \partial_th + u\partial_xh + v\partial_yh = -(H + \eta - z_B)(\partial_xu + \partial_yu) \] | |||
1.5-Layer SWE | \[ \partial_tu + u\partial_xu + v\partial_yu - fv = -g \left( \frac{\rho_2 - \rho_1}{\rho_1} \right) \partial_xz_2 \] \[ \partial_tv + u\partial_xv + v\partial_yv + fu = -g \left( \frac{\rho_2 - \rho_1}{\rho_1} \right) \partial_xz_2 \] \[ \partial_th + u\partial_xh + v\partial_yh = -h(\partial_xu + \partial_yu) \] | |||
2-Layer SWE | \[ \partial_tu + u\partial_xu + v\partial_yu - fv = -g\partial_xz_1 - g \left( \frac{\rho_2 - \rho_1}{\rho_1} \right) \partial_xz_2 \] \[ \partial_tv + u\partial_xv + v\partial_yv + fu = -g\partial_yz_1 - g \left( \frac{\rho_2 - \rho_1}{\rho_1} \right) \partial_yz_2 \] \[ \partial_th + u\partial_xh + v\partial_yh = -h(\partial_xu + \partial_yu) \] | |||
Linearized 1-Layer SWE | \[ \partial_tu' - f_0v = -g\partial_x\eta' \] \[ \partial_tv' + f_0u = -g\partial_y\eta' \] \[ \partial_t\eta' + H(\partial_xu' + \partial_yu') = 0 \] | |||
Dispersion Relation | \[ \omega [f_0^2 - \omega^2 + c_0^2 \kappa^2 ] \eta_0 = 0 \] \[ \omega_1 = 0 \] \[ \omega_{2,3}^2 = f_0^2 + c_0^2 \kappa^2 \] | |||
Rossby Dispersion Relation | \[ \omega = \bar{u}k - \frac{\beta k}{ \kappa^2} \ \ \ (Non-Divergent) \] \[ \omega = k \left( \bar{u} - \frac{\beta + \frac{\bar{u}}{L_D^2} }{ \kappa^2 + L_D^{-2}} \right) \ \ \ (Divergent) \] | |||
Deformation Radius | \[ L_D = \frac{c_0}{f_0} = \frac{\sqrt{gH}}{f_0} \] | |||
Length Scale | \[ L = \frac{1}{\kappa^2} \] | |||
Velocity Potential | \[ \vec{U} = \nabla \cdot \phi = \left \langle \frac{\partial \phi}{\partial x},\ \frac{\partial \phi}{\partial y},\ 0 \right \rangle \] | |||
Streamfunction | \[ \vec{U} = \nabla \times \psi = \left \langle -\frac{\partial \psi}{\partial y},\ \frac{\partial \psi}{\partial x},\ 0 \right \rangle \] \[ \vec{\omega} = \nabla^2 \psi = \nabla \times \vec{U} \] \[ \psi = \frac{gz}{f} = \frac{\Phi}{f} \] | |||
Vorticity | \[ \vec{\omega} = \nabla \times \vec{U} = 2\vec{\Omega} = \left \langle \xi,\ \eta,\ \zeta \right \rangle \] \[ \zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \] \[ \zeta_{a} = \zeta + f \] | |||
Full Vorticity Equation | \begin{eqnarray} \frac{d \zeta_a}{dt} &=& -(\zeta + f) \vec{\nabla} \cdot \vec{U} + \left( \frac{\partial u}{\partial z}\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}\frac{\partial w}{\partial x} \right) + \frac{ \nabla \rho \times \nabla p}{\rho^2} + \nu \nabla^2 \zeta \end{eqnarray} Vorticity Change = Stretching + Tilting + Baroclinic + Diffusion | |||
Baroclinic | \[ \rho \neq \rho(p) \] | |||
Barotropic | \[ \rho = \rho(p) \] | |||
Non-Divergent BVE | \[ \frac{d}{dt}(\zeta + f) = 0 \] | |||
Divergent BVE | \[ \frac{d}{dt}(\zeta + f) = -(\zeta + f) \vec{\nabla} \cdot \vec{U} + \vec{\nabla} \times \vec{F} \] | |||
Frontogenesis Equation | \begin{eqnarray} F &=& \frac{\partial u}{\partial y}\frac{\partial \theta}{\partial x} + \frac{\partial v}{\partial y}\frac{\partial \theta}{\partial y} + \frac{\partial \omega}{\partial y}\frac{\partial \theta}{\partial p} - \frac{\partial}{\partial y} \left( \frac{\theta}{T}\frac{Q}{c_p} \right) \end{eqnarray} F = Shearing + Stretching + Tilting + Heating | |||
Quasi-Geostrophic Approximation
Non-Dimensional SWE |
\[ \varepsilon \left( \partial_tu + u\partial_xu + v\partial_yu \right) -
\left( 1 + \gamma\varepsilon y \right)v = -\partial_x \eta + \varepsilon E_x \]
\[ \varepsilon \left( \partial_tv + u\partial_xv + v\partial_yv \right) +
\left( 1 + \gamma\varepsilon y \right)u = -\partial_y \eta + \varepsilon E_y \]
\[ \varepsilon F \left[ \partial_t \eta + u\partial_x (\eta - z_B) +
v\partial_y (\eta - z_B) \right] + \left[ 1 + \varepsilon F (\eta - z_B) \right] \left( \partial_xu + \partial_yv \right) \]
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SWPV Equation | \[ \frac{d}{dt} \left(\frac{\zeta + f}{h} \right) = \frac{\vec{\nabla} \times \vec{F}}{h} \simeq 0 \] | ||
QGPV Equation | \[ \frac{D_g}{Dt}q = \partial_tq + J(\psi, q) = \vec{\nabla} \times \left(\frac{\vec{F}}{\rho}\right) \] \[ q = f_0 + \beta y + \nabla^2 \psi - \frac{\psi}{L^2_D} + \frac{f_0}{D}z_B \] | ||
Linearized QGPV | \[ (\partial_t + U\partial_x)q' + \bar{\beta} \partial_x \psi' = 0 \] \[ q' = \nabla^2 \psi' - \frac{\psi'}{L^2_D} \] \[ \bar{\beta} = \beta + \frac{U}{L^2_D} \] | ||
QG Rossby Waves | \[ \omega = Uk - \frac{\bar{\beta}k}{\kappa^2 + L^{-2}_D} \] | ||
QG Momentum Equations | \[ \frac{d_gu_g}{dt} - f_0v_a - \beta y v_g = 0 \] \[ \frac{d_gv_g}{dt} + f_0u_a + \beta y u_g = 0 \] | ||
QG Vorticity | \[ \zeta_g = \vec{k} \cdot \nabla \times \vec{U_g} = \frac{\nabla^2 \psi}{f_0} \] | ||
QG Vorticity Equation | \[ \frac{\partial}{\partial t} \left( \zeta_g + f \right) = -\vec{U_g} \cdot \nabla \left( \zeta_g + f \right) + f_0 \frac{\partial \omega}{\partial p} \] Local Vort Change = Horiz Vort Advection + Stretching | ||
QG Thermo Equation | \[ \frac{d_gT}{dt} = -\omega \left( \frac{T_0}{\theta_0} \frac{\partial \theta_0}{\partial p} \right) + \frac{J}{c_p} \] Temp Change = Adiabatic Heating + Diabatic Heating | ||
Tendency Equation | \begin{eqnarray} \left( \nabla^2 + \frac{f^2_0}{\sigma}\frac{\partial^2}{\partial p^2} \right) \frac{\partial \Phi}{\partial t} &=& -f_0 \vec{V_g} \cdot \vec{\nabla} \left( \frac{1}{f_0} \nabla^2 \Phi + f \right) + \frac{f^2_0}{\sigma} \frac{\partial}{\partial p} \left[ \vec{V_g} \cdot \nabla \left( \frac{\partial \Phi}{\partial p} \right) \right] \end{eqnarray} -Height Tendency = Vort Advection + Differential Temp Advection | ||
Omega Equation | \begin{eqnarray} \left( \nabla^2 + \frac{f^2_0}{\sigma}\frac{\partial^2}{\partial p^2} \right) \omega &=& \frac{f_0}{\sigma}\frac{\partial}{\partial p} \left[ \vec{V_g} \cdot \vec{\nabla} \left( \frac{1}{f_0} \nabla^2 \Phi + f \right) \right] + \frac{1}{\sigma} \nabla^2 \left[ \vec{V_g} \cdot \vec{\nabla} \left( -\frac{\partial \Phi}{\partial p} \right) \right] \end{eqnarray} -Omega = Vort Advection + Differential Temp Advection | ||
Advanced Dynamics
Shallow Water Equations | \[ \partial_tu + u\partial_xu + v\partial_yu - fv = -g\partial_x\eta + F_x \] \[ \partial_tv + u\partial_xv + v\partial_yv + fu = -g\partial_y\eta + F_y \] \[ \partial_th + u\partial_xh + v\partial_yh + h(\partial_xu + \partial_yu) = 0 \] \[ h = \eta - z_B \] | ||
1.5-Layer Model | \[ \partial_tu + u\partial_xu + v\partial_yu - fv = -g'\partial_xh + F_x \] \[ \partial_tv + u\partial_xv + v\partial_yv + fu = -g'\partial_yh + F_y \] \[ \partial_th + u\partial_xh + v\partial_yh + h(\partial_xu + \partial_yu) = 0 \] \[ h \approx - z_B \] | ||
Reduced Gravity | \[ g' = \frac{\rho_2 - \rho_1}{\rho_1}g \] | ||
Linearized SW on f-Plane | \[ \partial_tu' - f_0v = -g\partial_x\eta' \] \[ \partial_tv' + f_0u = -g\partial_y\eta' \] \[ \partial_t\eta' + H(\partial_xu' + \partial_yu') = 0 \] | ||
Linearized Variables | \[ \eta = H + \eta' \] \[ u = u' \] \[ v = v' \] | ||
Eigenvalues (3 Eigenmodes) |
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Reynold's Stress | \[ \tau = -\overline{u'w'} = k\frac{\partial U}{\partial z} \] | ||
Ekman Transport | \[ U_E = \frac{\tau^y}{f\rho} \] \[ V_E = -\frac{\tau^x}{f\rho} \] | ||
Ekman Pumping | \[ w_E = \frac{1}{f\rho} \vec{\nabla} \times \vec{\tau} \] | ||
Sverdrup Flow | \[ \beta V_s = \vec{\nabla} \times \left( \frac{ \vec{\tau} }{\rho} \right) \] \[ \psi_s = \int_{x_E}^{x} \frac{\vec{\nabla} \times \vec{\tau}}{\rho}dx \] \[ \psi_s = \int_{x_W}^{x} \frac{\vec{\nabla} \times \vec{\tau}}{\rho}dx \] | ||