🌊 Physical Oceanography Equations

Geostrophic Equations:

$$-fv_g = -\frac{1}{\rho}\frac{\partial p}{\partial x}$$

$$fu_g = -\frac{1}{\rho}\frac{\partial p}{\partial y}$$

In terms of sea surface height:

$$u_g = -\frac{g}{f}\frac{\partial \eta}{\partial y}$$

$$v_g = \frac{g}{f}\frac{\partial \eta}{\partial x}$$

Thermal Wind Equations:

$$\frac{\partial u_g}{\partial z} = -\frac{g}{\rho_0 f}\frac{\partial \rho}{\partial y}$$

$$\frac{\partial v_g}{\partial z} = \frac{g}{\rho_0 f}\frac{\partial \rho}{\partial x}$$

Integrated form:

$$\mathbf{V}_g(z_2) - \mathbf{V}_g(z_1) = \frac{g}{\rho_0 f}\int_{z_1}^{z_2} \mathbf{k} \times \nabla_h \rho \, dz$$

Ekman Balance:

$$-fv = A_z\frac{\partial^2 u}{\partial z^2}$$

$$fu = A_z\frac{\partial^2 v}{\partial z^2}$$

Ekman Solution:

$$u(z) = V_0 e^{z/D_E} \cos\left(\frac{\pi}{4} + \frac{z}{D_E}\right)$$

$$v(z) = V_0 e^{z/D_E} \sin\left(\frac{\pi}{4} + \frac{z}{D_E}\right)$$

Ekman Depth: $D_E = \sqrt{\frac{2A_z}{f}}$

Ekman Pumping Velocity:

$$w_E = \frac{1}{\rho f}\left(\frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y}\right)$$

$$w_E = \frac{\text{curl}(\boldsymbol{\tau})}{\rho f}$$

Sverdrup Transport:

$$\beta \int_{x_e}^{x_w} v \, dx = \frac{\text{curl}(\boldsymbol{\tau})}{\rho}$$

Dispersion Relation:

$$\omega^2 = gk \tanh(kh)$$

Deep Water Waves (kh >> 1):

$$\omega^2 = gk, \quad c = \sqrt{\frac{g}{k}} = \sqrt{\frac{gT}{2\pi}}$$

Shallow Water Waves (kh << 1):

$$\omega^2 = gk^2h, \quad c = \sqrt{gh}$$

Group Velocity:

$$c_g = \frac{\partial \omega}{\partial k} = \frac{1}{2}c\left(1 + \frac{2kh}{\sinh(2kh)}\right)$$

Brunt-Väisälä Frequency:

$$N^2 = -\frac{g}{\rho_0}\frac{d\rho}{dz} = \frac{g}{\rho_0}\left(\alpha\frac{dT}{dz} - \beta\frac{dS}{dz}\right)$$

Internal Wave Dispersion:

$$\omega^2 = N^2\frac{k_h^2}{k_h^2 + k_z^2} + f^2\frac{k_z^2}{k_h^2 + k_z^2}$$

Inertial-Gravity Waves:

$$f^2 \leq \omega^2 \leq N^2$$

Temperature Equation:

$$\frac{DT}{Dt} = \kappa_T \nabla^2 T + \frac{Q}{\rho c_p}$$

Salinity Equation:

$$\frac{DS}{Dt} = \kappa_S \nabla^2 S + \frac{S(E-P)}{\rho}$$

Potential Temperature:

$$\theta = T - \int_0^p \frac{\alpha T}{c_p} dp'$$

Potential Density:

$$\sigma_\theta = \rho(\theta, S, 0) - 1000 \text{ kg/m}^3$$

Meridional Overturning Circulation:

$$\Psi(y,z) = \int_{z}^{0} \int_{x_w}^{x_e} v(x,y,z') \, dx \, dz'$$

Stommel's Box Model:

$$\frac{dT_1}{dt} = \gamma(T_0 - T_1) - q(T_1 - T_2)$$

$$\frac{dS_1}{dt} = \gamma(S_0 - S_1) - q(S_1 - S_2)$$

$$q = k|\alpha(T_1-T_2) - \beta(S_1-S_2)|$$

Reynolds Decomposition:

$$u = \bar{u} + u', \quad v = \bar{v} + v', \quad w = \bar{w} + w'$$

Turbulent Kinetic Energy:

$$\text{TKE} = \frac{1}{2}(\overline{u'^2} + \overline{v'^2} + \overline{w'^2})$$

Richardson Number:

$$Ri = \frac{N^2}{(\partial u/\partial z)^2 + (\partial v/\partial z)^2}$$

Critical Richardson Number: $Ri_c = 0.25$

Osborn Model:

$$K_\rho = \Gamma \frac{\varepsilon}{N^2}$$

Where: $\Gamma = 0.2$ (mixing efficiency)

Thorpe Scale:

$$L_T = \sqrt{\overline{d^2}}$$

Ozmidov Scale:

$$L_O = \sqrt{\frac{\varepsilon}{N^3}}$$

Laplace Tidal Equations:

$$\frac{\partial u}{\partial t} - fv = -g\frac{\partial \eta}{\partial x} + \frac{\tau_x}{\rho H}$$

$$\frac{\partial v}{\partial t} + fu = -g\frac{\partial \eta}{\partial y} + \frac{\tau_y}{\rho H}$$

$$\frac{\partial \eta}{\partial t} + \frac{\partial}{\partial x}(Hu) + \frac{\partial}{\partial y}(Hv) = 0$$

Tidal Potential:

$$V = \sum_{n,m} A_{nm} P_n^m(\sin\phi) \cos(m\lambda + \omega t + \phi_{nm})$$