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Geophysics operators

61 operators in the geophysics category of the live registry. Each is a named formula you can compose inside a state contract or call directly through POST /api/zeq/compute. KO42 is always on; add up to three more per call (total ≤ 4), per the 7-step protocol.

OperatorDescriptionEquation
GP1Gravitational acceleration at distance r from a point mass, derived from Newton's universal law of gravitation.g = \frac{GM}{r^2}
GP10Moment magnitude with 1.287 Hz kinematic modulation for Zeq-enhanced seismic magnitude estimation.M_w = \frac{2}{3}(\log_{10}M_0 - 9.1) \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
GP11Fourier's law of heat conduction stating that heat flux is proportional to the negative temperature gradient scaled by thermal conductivity.q = -k\nabla T
GP12Geothermal gradient expressing the rate of temperature increase with depth as the ratio of heat flux to thermal conductivity.\frac{\partial T}{\partial z} = \frac{q}{k}
GP13Thermal density variation approximation relating density change to temperature departure from a reference state through the thermal expansion coefficient.\rho = \rho_0(1 + \alpha\Delta T)
GP14Hydrostatic pressure equation giving the pressure at depth in a fluid column from density, gravitational acceleration, and height.P = \rho g h
GP15Poisson's equation for gravitational potential relating its Laplacian to the local mass density scaled by the gravitational constant.\nabla^2 U = -4\pi G\rho
GP16Geoid height computed as the ratio of the disturbing potential to normal gravity, measuring the geoid-ellipsoid separation.N = \frac{U}{g}
GP17North-south component of the deflection of the vertical, measuring the angular difference between the geoid normal and the ellipsoid normal.\xi = \frac{1}{R}\frac{\partial N}{\partial\phi}
GP18East-west component of the deflection of the vertical, measuring the angular deviation of the plumb line from the geodetic normal.\eta = \frac{1}{R\cos\phi}\frac{\partial N}{\partial\lambda}
GP19Rayleigh number indicating the onset of thermal convection as the ratio of buoyancy-driven to diffusion-restrained transport in a fluid layer.Ra = \frac{g\alpha\Delta T d^3}{\nu\kappa}
GP2Gravity anomaly defined as the difference between observed and reference gravitational acceleration, revealing subsurface density variations.\Delta g = g_{obs} - g_{ref}
GP20Peclet number comparing advective to diffusive heat transport, governing whether convection or conduction dominates thermal transfer.Pe = \frac{vL}{\kappa}
GP3Period of a physical (compound) pendulum determined by the moment of inertia, mass, and distance from pivot to center of gravity.T = 2\pi\sqrt{\frac{I}{mgh}}
GP4Magnetic field relation in matter combining the applied field and magnetization through the permeability of free space.B = \mu_0(H + M)
GP5Zeq-modified Ampere's law incorporating a 1.287 Hz sinusoidal modulation into the current density for geomagnetic field modeling.\nabla \times \vec{B} = \mu_0 \vec{J} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
GP6Gauss's law for electric displacement relating the divergence of the D-field to the free charge density in a dielectric medium.\nabla \cdot \vec{D} = \rho_f
GP7P-wave velocity with 1.287 Hz kinematic modulation for Zeq geophysical wave propagation analysis.v_P = \sqrt{\frac{K + 4\mu/3}{\rho}} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
GP8S-wave velocity with 1.287 Hz kinematic modulation for Zeq geophysical shear wave propagation analysis.v_S = \sqrt{\frac{\mu}{\rho}} \cdot [1 + \alpha \sin(2\pi \cdot 1.287t)]
GP9Local magnitude scale computing earthquake size from the logarithm of maximum seismogram amplitude with distance corrections.M_L = \log_{10}A - \log_{10}A_0
GPO1Plate tectonic velocity describing the rate and direction of lithospheric plate motion relative to a fixed reference frame.\vec{v}{plate} = \vec{\omega} \times \vec{r} + \vec{v}{mantle} + \vec{v}{slab_pull} + \vec{v}{ridge_push}
GPO10Subduction zone thermal structure modeling the temperature field of a descending slab from plate age, velocity, and dip angle.v_{slab} = \sqrt{\frac{\Delta \rho g \sin\theta}{\rho \eta}} L \cdot \eta_{rheology} \cdot f_{age}
GPO11Continental rift valley extension rate relating crustal thinning to far-field tectonic stress and lithospheric rheology.\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\left(D\frac{\partial u}{\partial x}\right) + \dot{\epsilon}_0 + \alpha \Delta T + \beta \nabla^2 u
GPO12Hotspot track model predicting the age-distance progression of volcanic islands from plate motion over a stationary mantle plume.\frac{d\vec{x}{volcano}}{dt} = \vec{v}{plate} - \vec{v}{plume} + \vec{v}{mantle_wind} + \vec{\eta}_{deflection}
GPO2Mantle convection equation coupling thermal buoyancy with viscous flow to model large-scale circulation in the Earth's mantle.\frac{D\vec{v}}{Dt} = -\frac{1}{\rho}\nabla p + \nu\nabla^2\vec{v} + \alpha g \Delta T \hat{r} + \vec{F}_{compositional}
GPO3Terrestrial heat flow measured as the product of thermal conductivity and the vertical temperature gradient through the lithosphere.q = -k\nabla T + \rho c_p \vec{v} T + q_{radioactive} + q_{adiabatic}
GPO4Isostatic equilibrium principle balancing crustal loads against buoyant support from the underlying mantle to determine topographic compensation.\rho_c h_c = \rho_m h_m, \quad \Delta h = \frac{\rho_m - \rho_l}{\rho_m} h_l
GPO5Fault mechanics relating shear stress, normal stress, and friction coefficient to predict slip initiation on geological faults.\tau = \tau_0 + \mu(\sigma_n - p) + A \ln\left(\frac{V}{V_0}\right) + B \ln\left(\frac{V_0 \theta}{D_c}\right)
GPO6Earthquake scaling law relating rupture dimensions, stress drop, and seismic moment for estimating expected ground motion.M_0 = \mu A D, \quad M_w = \frac{2}{3}\log_{10} M_0 - 6.07
GPO7Seismic wave propagation equation describing how elastic waves travel through heterogeneous Earth materials with velocity and attenuation.\rho \frac{\partial^2 u_i}{\partial t^2} = \frac{\partial \sigma_{ij}}{\partial x_j} + f_i, \quad \sigma_{ij} = C_{ijkl} \epsilon_{kl}
GPO8Volcanic eruption dynamics modeling magma ascent rate from chamber overpressure, conduit geometry, and magma viscosity.Q = \frac{\pi \Delta P r^4}{8\mu L} \cdot f(\phi) \cdot g(Re), \quad \Delta P = \rho g \Delta h - \Delta P_{viscous}
GPO9Magma chamber pressure evolution from the balance of magma influx, volatile exsolution, and elastic wall rock deformation.\frac{dV}{dt} = Q_{in} - Q_{out} - \frac{dV_{crystallization}}{dt} + \beta V \Delta T
MET1Hydrostatic equation relating the vertical pressure gradient to the product of air density and gravitational acceleration in the atmosphere.\frac{dp}{dz} = -\rho g
MET10Potential vorticity combining absolute vorticity and static stability into a conserved quantity for adiabatic, frictionless flow.\zeta = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}
MET11Omega equation diagnosing vertical velocity in pressure coordinates from differential vorticity advection and thermal advection.\frac{d\zeta}{dt} = -(\zeta + f)\nabla \cdot \vec{v}
MET12Convective Available Potential Energy integrating buoyancy of a lifted parcel over depth to assess thunderstorm potential.PV = \frac{\zeta + f}{\rho}\frac{\partial\theta}{\partial z}
MET13Atmospheric kinetic energy per unit mass computed from wind speed squared, used in storm intensity and energy budget analyses.\omega = \frac{dp}{dt}
MET14Atmospheric moisture flux convergence driving precipitation, computed from the divergence of the vertically integrated water vapor transport.CAPE = \int_{LFC}^{EL} g\frac{T_p - T_e}{T_e} dz
MET15Atmospheric radiation balance determining net radiative heating from the difference between absorbed shortwave and emitted longwave radiation.K = \frac{1}{2}(u^2 + v^2)
MET2Barometric formula predicting exponential pressure decrease with altitude in an isothermal atmosphere using the scale height.P = P_0 e^{-z/H}
MET3Environmental lapse rate giving the rate of temperature decrease with altitude in the troposphere under standard conditions.\Gamma = -\frac{dT}{dz} = 9.8 \text{ K/km}
MET4Potential temperature converting measured temperature to the value it would have if adiabatically brought to a reference pressure level.\Gamma_s = \Gamma\frac{1 + L_v r_s / R_d T}{1 + L_v^2 r_s / c_p R_v T^2}
MET5Clausius-Clapeyron equation describing the exponential increase of saturation vapor pressure with temperature for phase equilibria.\theta = T\left(\frac{P_0}{P}\right)^{R/c_p}
MET6Relative humidity expressed as the ratio of actual water vapor pressure to the saturation vapor pressure at the same temperature.e_s = 6.11 \exp\left(\frac{17.27 T}{T + 237.3}\right)
MET7Geostrophic wind speed resulting from the balance between the pressure gradient force and the Coriolis force in large-scale flow.RH = \frac{e}{e_s} \times 100\%
MET8Ageostrophic wind component representing the departure from geostrophic balance that drives vertical motion and frontogenesis.\vec{v}_g = -\frac{1}{\rho f}\hat{k} \times \nabla P
MET9Relative vorticity measuring the local spin of air parcels about a vertical axis from the horizontal wind field derivatives.\vec{v}_a = \vec{v} - \vec{v}_g
OC10Richardson number comparing buoyancy-driven stability to velocity shear, predicting the onset of turbulent mixing in stratified flows.Ri = \frac{N^2}{(\partial u/\partial z)^2}
OC11Brunt-Vaisala frequency measuring the natural oscillation frequency of a vertically displaced fluid parcel in a stably stratified ocean.N = \sqrt{-\frac{g}{\rho}\frac{\partial\rho}{\partial z}}
OC12Seawater equation of state relating density to temperature, salinity, and pressure for computing ocean buoyancy and circulation.\rho = \rho(S, T, P)
OC13Streamfunction for incompressible flow defining velocity components such that mass conservation is automatically satisfied.\Psi = \int \vec{u} \cdot d\vec{A}
OC14Volume transport through an ocean section computed by integrating current velocity over the cross-sectional area.Q = \int\int \vec{u} \cdot \hat{n} dA
OC15Tidal harmonic analysis decomposing sea level variations into constituent sinusoidal components with known astronomical frequencies.\eta_{tide} = \sum_i H_i\cos(\omega_i t + \phi_i)
OC2Deep-water surface gravity wave phase speed depending on wavelength and gravitational acceleration through the dispersion relation.c = \sqrt{\frac{g\lambda}{2\pi}\tanh\frac{2\pi H}{\lambda}}
OC3Wave breaking height criterion estimating the maximum stable wave height as a fraction of the local water depth.H_b = 0.78 H_s
OC4Shallow water wave equations governing long-wave propagation where wave speed depends on the square root of depth times gravity.\frac{\partial\eta}{\partial t} + \nabla \cdot (H\vec{u}) = 0
OC5Ekman transport giving the net horizontal mass transport in the wind-driven ocean surface layer directed perpendicular to the wind stress.\vec{u}_E = \frac{\vec{\tau}}{\rho f D_E}
OC6Ekman layer depth estimating the thickness of the wind-driven surface layer from vertical eddy viscosity and the Coriolis parameter.D_E = \sqrt{\frac{2A_v}{f}}
OC7Coriolis parameter as a function of latitude, quantifying the deflective effect of Earth's rotation on moving fluid parcels.f = 2\Omega\sin\phi
OC8Geostrophic current velocity derived from the balance between the horizontal pressure gradient and the Coriolis force in the ocean.\vec{u}_g = \frac{1}{\rho f}\hat{k} \times \nabla P
OC9Rossby number comparing inertial to Coriolis forces, indicating whether rotation or advection dominates the flow dynamics.Ro = \frac{U}{fL}

Compute with one of these​

curl -sS -X POST https://zeqsdk.com/api/zeq/compute \
  -H "Authorization: Bearer $ZEQ_KEY" \
  -H "Content-Type: application/json" \
  -d '{"operators":["GP1"],"inputs":{}}'

The response carries the bare physics value, its unit and uncertainty, the generated master equation, and a signed envelope you can verify on any node.

See also​