Foundations

For a constant-width reach with constant bed porosity p, sediment-solid conservation gives (1−p)Bη_t=−∂Q_s/∂x.

An isotropic incompressible Glen-type model relates deviatoric stress τ to strain rate by ε̇ᵢⱼ=Aτ_e^(n−1)τᵢⱼ, where τ_e=√(τ:τ/2),A>0 and n>0.

For effectively constant-density ice, depth-integrated mass conservation is H_t+∇_h·(Hū)=a_s−m_b.

Phanerozoic reconstructions include the late Paleozoic assembly of Pangaea, Mesozoic rifting and opening of major ocean basins, and later collisions and accretions that reorganized continental margins.

For constant pumping from an ideal confined aquifer, drawdown at radius r and time t is s=QW(u)/(4πT), where u=r²S/(4Tt) and W(u)=∫ᵤ∞e^(−ξ)dξ/ξ.

Hydrostatic and nearly geostrophic balances imply vertical geostrophic shear from horizontal density gradients: ∂u_g/∂z=gρ_y/(fρ₀), ∂v_g/∂z=−gρ_x/(fρ₀).

For a dry ideal-gas atmosphere in hydrostatic/geostrophic balance, pressure-coordinate shear satisfies ∂u_g/∂p=R_d T_y/(fp) and ∂v_g/∂p=−R_d T_x/(fp).

For two uniform-density layers separated by z_i(y), pressure continuity across the interface relates its slope to along-front geostrophic velocities: dz_i/dy=f(ρ₁u₁−ρ₂u₂)/[g(ρ₂−ρ₁)].

For an inviscid uniform-density layer over a flat fixed bottom, depth-averaged motion obeys u_t+u·∇_hu+fe_z×u=−g∇_hh and h_t+∇_h·(hu)=0.

In a smooth inviscid single-layer shallow-water model without mass exchange or external torque, q=(ζ+f)/h is materially conserved: Dq/Dt=0, with ζ=∂v/∂x−∂u/∂y.

Linearizing a uniform-depth rotating shallow-water layer at rest on an f-plane yields propagating roots ω²=f²+gH(k_x²+k_y²) and a balanced zero-frequency mode.

For small-amplitude low-frequency nearly geostrophic perturbations of a resting uniform layer, f=f₀+βy gives planetary Rossby-wave dispersion ω=−βk_x/(k_x²+k_y²+L_d⁻²).

In a constant-density,constant-heat-capacity incompressible sensible-heat model, ρ₀c_pDT/Dt=−∇·J_H+q_H.

With salt mass fraction s, conservation is ∂(ρs)/∂t+∇·(ρsu+J_s)=r_s.

A well-mixed layer of fixed depth H with constantρ₀,c_p and negligible lateral/entrainment heat transport has ρ₀c_pH dT/dt=Q_net,where Q_net is the total net heat input per horizontal area.

For a well-mixed reservoir with solution mass M,positive salt fraction s and conserved salt mass Ms, freshwater gain/loss gives ds/dt=−(s/M)dM/dt.

Differential lunar and solar gravity supplies tidal forcing, but ocean height and current respond through rotating fluid dynamics,bathymetry,coastlines and dissipation.

A bounded harmonic model represents astronomical water-level tide as η(t)=η₀+Σ_jA_jcos(ω_jt+φ_j).

Changes in orbital eccentricity,obliquity and the orientation of perihelion relative to the seasons alter incoming solar energy's seasonal and latitudinal distribution.

For solar flux S perpendicular to the rays at the current orbital distance,latitudeφ and declinationδ give cosZ=sinφsinδ+cosφcosδcosh.

In a small-displacement incompressible Boussinesq parcel model, N²=−(g/ρ₀)dρ_b/dz gives the vertical restoring acceleration ξ̈=−N²ξ when parcel density is conserved.

For a resting continuously stratified incompressible Boussinesq fluid with constant stable N and constant f,linear plane-wave internal modes obey ω²=(N²k_h²+f²m²)/(k_h²+m²).

For steady inviscid incompressible homogeneous fluid in the leading small-Rossby rotating balance, taking the curl of 2Ω×u=−∇(p/ρ+Φ_eff) gives (Ω·∇)u=0.

Magma ascent can lower confining pressure enough for dissolved volatiles to exsolve and expand.

A tectonic earthquake involves sudden slip on a fault and release of part of the surrounding rock's stored elastic energy, including energy radiated as seismic waves.

In a compression-positive Coulomb model for a closed fault, effective normal traction is σ′_n=σ_n−p.

For a planar shear-fault source with uniform shear modulus μ and consistently directed slip magnitude D, scalar moment is M₀=μAD̄, equivalently μ∫_A D dA.

For a homogeneous isotropic linearly elastic solid of densityρ, P waves have vP²=(λ+2μ)/ρ=(K+4μ/3)/ρ and S waves have vS²=μ/ρ.

In a high-frequency ray model, travel time along a pathγ is t=∫_γ ds/v.

For source and receiver on a flat surface a horizontal distance x apart above a uniform layer of depth h and speed v₁, direct travel time is x/v₁ and reflection from its horizontal base has time 2√(h²+x²/4)/v₁.