📐 Advanced Governing Equations
Enhanced Slab Model with Complex Physics:
$$\frac{\partial u}{\partial t} - fv = \frac{\tau_x}{\rho h} - r_u u + \frac{\partial}{\partial t}\left(\frac{h_0 - h}{h}\right)u_0 - \alpha \frac{\partial h}{\partial t}u$$
$$\frac{\partial v}{\partial t} + fu = \frac{\tau_y}{\rho h} - r_v v + \frac{\partial}{\partial t}\left(\frac{h_0 - h}{h}\right)v_0 - \alpha \frac{\partial h}{\partial t}v$$
$$\frac{\partial h}{\partial t} = w_e - \frac{\nabla \cdot (\mathbf{U}h)}{A} + \beta(T_s - T_{ml})$$
Wind Stress with Rotation:
$$\tau_x = \rho_a C_d |\mathbf{W}| [W_u \cos(\theta) - W_v \sin(\theta)]$$
$$\tau_y = \rho_a C_d |\mathbf{W}| [W_u \sin(\theta) + W_v \cos(\theta)]$$
Ekman Pumping:
$$w_e = \frac{1}{\rho f} \left(\frac{\partial \tau_y}{\partial x} - \frac{\partial \tau_x}{\partial y}\right)$$
Parameters: $r_{u,v}$ = friction coefficients, $\alpha$ = entrainment parameter, $\theta$ = wind stress rotation angle, $w_e$ = Ekman pumping velocity