📐 Complete Physical Equations
Total Current Velocity:
U_total(t) = U_ekman(t) + U_geostrophic(t) + U_inertial(t) + U_tidal(t) + U_background
1. Wind Stress:
τ = ρ_air × C_d × |W| × W
where ρ_air = 1.225 kg/m³, C_d = 1.3×10⁻³, W = wind velocity
2. Ekman Current (Wind-Induced):
∂u/∂t - f×v = (1/ρ_w×h) × τ_x - r×u
∂v/∂t + f×u = (1/ρ_w×h) × τ_y - r×v
where f = 2Ω sin(φ), Ω = 7.2921×10⁻⁵ rad/s, φ = latitude, r = friction coefficient
3. Near-Inertial Oscillations:
U_inertial = A(t) × exp(-t/T_d) × [cos(f×t + φ₀), sin(f×t + φ₀)]
A(t) = α × |τ_wind|/f, T_d = 2-5 days, α = 0.15 m²s⁻²N⁻¹
4. Tidal Harmonics:
U_tidal = Σᵢ Aᵢ × [cos(ωᵢt + φᵢ), sin(ωᵢt + ψᵢ)]
M2: ω = 1.405×10⁻⁴ rad/s, S2: ω = 1.454×10⁻⁴ rad/s
K1: ω = 7.292×10⁻⁵ rad/s, O1: ω = 6.759×10⁻⁵ rad/s
5. Geostrophic Balance:
f×u_g = -g × ∂η/∂y, f×v_g = g × ∂η/∂x
where η = sea surface height, g = 9.81 m/s²
6. Background Circulation:
U_background = constant large-scale flow (boundary currents, gyres)
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