Oceanography
Posted by Alex
• • Full article
Properties of Water
Density of Fresh Water |
\[ \rho_{fresh} = 1000 \ \frac{kg}{m^3} \]
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Density of Seawater |
\[ \rho_{sea} = 1025 \ \frac{kg}{m^3} \]
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Density of Ice |
\[ \rho_{ice} = 917 \ \frac{kg}{m^3} \]
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Dynamic Viscosity |
\[ \mu_{water} = 10^{-3} \ \frac{kg}{m \ s} \]
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Kinematic Viscosity |
\[ \nu_{water} = \frac{\mu}{\rho} = 10^{-6} \ \frac{m^2}{s} \]
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Thermal Diffusivity |
\[ k = 1.4 \times 10^{-6} \ \frac{m^2}{s} \]
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Thermal Expansion Coefficient |
\[ \alpha_T = -\frac{1}{\rho_{ref}}\frac{\partial \rho}{\partial T} \simeq 1 \times 10^{-4} \ ^{\circ}{\rm C}^{-1} \]
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Haline Contraction Coefficient |
\[ \beta_S = \frac{1}{\rho_{ref}}\frac{\partial \rho}{\partial S} \simeq 7.6 \times 10^{-4} \ {psu}^{-1} \]
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Steric Effect |
\[ \frac{\Delta \eta}{H} = \left( \alpha_T \langle T-T_0 \rangle - \beta_S \langle S-S_0 \rangle \right) \]
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Wind-Driven Circulation
Wind Stress |
\[ \tau_{wind} = \rho_{air} c_D u_H^2 \]
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Reynold's Stress |
\[ \tau^x = -\overline{u'w'} = K_m \frac{\partial u}{\partial z} \]
\[ \tau^y = -\overline{v'w'} = K_m \frac{\partial v}{\partial z} \]
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Bulk Transfer Coefficient |
\[ c_D = 1.5 \times 10^{-3} \]
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Eddy Diffusion Coefficient |
\[ K_m \simeq 10^{-1} \ m^2 s^{-1} \]
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Molecular Viscosity Coefficient |
\[ \nu \simeq 10^{-6} \ m^2 s^{-1} \]
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Ekman Number |
\[E_k = \left( \frac{Viscous}{Coriolis} \right) \simeq \frac{ \frac{K_m U}{d^2} }{ fU } = \frac{K_m}{fd^2} \]
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Ekman Equations |
\[ -fv = K_m \frac{\partial^2 u}{\partial z^2} = \frac{1}{\rho} \frac{\partial \tau^x}{\partial z} \]
\[ fu = K_m \frac{\partial^2 v}{\partial z^2} = \frac{1}{\rho} \frac{\partial \tau^y}{\partial z} \]
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Frictional Boundary Layer |
\[ d = \sqrt{ \frac{2\nu}{f} } \simeq 0.1 \ m \]
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Ekman Depth |
\[ \delta_E = \sqrt{ \frac{2K_m}{f} } \simeq 100 \ m \]
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Ekman Transport |
\[ M_E^x = \int_{0}^{z} \rho v dz = -\frac{\tau_x}{f} \]
\[ M_E^y = \int_{0}^{z} \rho u dz = \frac{\tau_y}{f} \]
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Ekman Pumping |
\[ w_E = \frac{1}{f\rho} \vec{\nabla} \times \vec{\tau} \]
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Sverdrup Flow |
\[ \beta V_s = \vec{\nabla} \times \left( \frac{ \vec{\tau} }{\rho} \right) \]
\[ \psi_s = \int_{x_E}^{x} \frac{\vec{\nabla} \times \vec{\tau}}{\rho}dx \]
\[ \psi_s = \int_{x_W}^{x} \frac{\vec{\nabla} \times \vec{\tau}}{\rho}dx \]
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Global Energy Balance
Radiative Equilibrium
(Without Atmosphere) |
\[ \left( 1-A \right)S_0 \pi a^2 = 4\pi a^2 \sigma T_s^4 \]
\[ T_s = \left[ \frac{S_0 \left( 1-A \right) }{4\sigma} \right]^{1/4} \simeq 255 \ {\rm K} \]
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Radiative Equilibrium
(With Atmosphere) |
\[ -\frac{S_0}{4} + A\left( 1-\alpha_{sw} \right)^2\frac{S_0}{4} + \sigma T_s^4 \left( 1-\alpha_{lw} \right) + \alpha_{lw} \sigma T_a^4 =0 \]
\[ -\left( 1-\alpha_{sw} \right) \frac{S_0}{4} + A\left( 1-\alpha_{sw} \right) \frac{S_0}{4} + \sigma T_s^4 - \alpha_{lw} \sigma T_a^4 = 0 \]
\[ T_s = \left[ \frac{S_0}{4\sigma} \left( \frac{2-\alpha_{sw}}{2-\alpha_{lw}} \right) \right]^{1/4} \simeq 286 \ {\rm K} \]
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Shortwave Absorptivity |
\[ \alpha_{sw} = 0.1 \]
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Longwave Absorptivity |
\[ \alpha_{lw} = 0.8 \]
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Climate Trends
Global Average Temperature |
\[ \Delta T \simeq 0.2 \ ^{\circ}{\rm C} \ {decade}^{-1} \ \ (since \ 1990) \]
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Sea Level Rise |
\[ \Delta H \simeq 3 \ mm \ {yr}^{-1} \ \ (since \ 1990) \]
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Winter Arctic Sea Ice Extent |
\[ \Delta E \simeq -3 \ \% \ {yr}^{-1} \ \ (since \ 1980) \]
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