Misplaced Pages

In gravity and pressure driven fluid dynamical and geophysical mass flows such as ocean waves, avalanches, debris flows, mud flows, flash floods, etc., kinematic waves are important mathematical tools to understand the basic features of the associated wave phenomena. These waves are also applied to model the motion of highway traffic flows.

In these flows, mass and momentum equations can be combined to yield a kinematic wave equation. Depending on the flow configurations, the kinematic wave can be linear or non-linear, which depends on whether the wave phase speed is a constant or a variable. Kinematic wave can be described by a simple partial differential equation with a single unknown field variable (e.g., the flow or wave height, ${\displaystyle h}$) in terms of the two independent variables, namely the time (${\displaystyle t}$) and the space (${\displaystyle x}$) with some parameters (coefficients) containing information about the physics and geometry of the flow. In general, the wave can be advecting and diffusing. However, in simple situations, the kinematic wave is mainly advecting.

Kinematic wave for debris flow

Non-linear kinematic wave for debris flow can be written as follows with complex non-linear coefficients: $${\displaystyle {\frac {\partial h}{\partial t}}+C{\frac {\partial h}{\partial x}}=D{\frac {\partial ^{2}h}{\partial x^{2}}},}$$ where ${\displaystyle h}$ is the debris flow height, ${\displaystyle t}$ is the time, ${\displaystyle x}$ is the downstream channel position, ${\displaystyle C}$ is the pressure gradient and the depth dependent nonlinear variable wave speed, and ${\displaystyle D}$ is a flow height and pressure gradient dependent variable diffusion term. This equation can also be written in the conservative form:

$${\displaystyle {\frac {\partial h}{\partial t}}+{\frac {\partial F}{\partial x}}=0,}$$

where ${\displaystyle F}$ is the generalized flux that depends on several physical and geometrical parameters of the flow, flow height and the hydraulic pressure gradient. For ${\displaystyle F=h^{2}/2}$, this equation reduces to the Burgers' equation.

References

1. Takahashi, T. (2007). Debris Flow: Mechanics, Prediction and Countermeasures. Taylor and Francis, Leiden.
2. Lighthill, M.J.; Whitham, G.B. (1955). "On kinematic waves. I: Flood movement in long rivers. II: A theory of traffic flow on long crowded roads". Proceedings of the Royal Society. 229A (4): 281–345.
3. Newell, G.F. (1993). "A simplified theory of kinematic waves in highway traffic, Part I: General theory". Transpn. Res. B. 27B (4): 281–287. Bibcode:1993TRPB...27..281N. doi:10.1016/0191-2615(93)90038-C.

Further reading

• Singh, Vijay P. (1996). "Linearization of Hydraulic Equations". Kinematic Wave Modeling in Water Resources : Surface-Water Hydrology. New York: John Wiley & Sons. pp. 211–253. ISBN 0-471-10945-2.

v t e Physical oceanography
Waves	Airy wave theory Ballantine scale Benjamin–Feir instability Boussinesq approximation Breaking wave Clapotis Cnoidal wave Cross sea Dispersion Edge wave Equatorial waves Gravity wave Green's law Infragravity wave Internal wave Iribarren number Kelvin wave Kinematic wave Longshore drift Luke's variational principle Mild-slope equation Radiation stress Rogue wave Rossby wave Rossby-gravity waves Sea state Seiche Significant wave height Soliton Stokes drift Stokes problem Stokes wave Swell Trochoidal wave Tsunami megatsunami Undertow Ursell number Wave action Wave base Wave height Wave nonlinearity Wave power Wave radar Wave setup Wave shoaling Wave turbulence Wave–current interaction Waves and shallow water one-dimensional Saint-Venant equations shallow water equations Wind fetch Wind setup Wind wave model
Circulation	Atmospheric circulation Baroclinity Boundary current Coriolis force Coriolis–Stokes force Craik–Leibovich vortex force Downwelling Eddy Ekman layer Ekman spiral Ekman transport El Niño–Southern Oscillation General circulation model Geochemical Ocean Sections Study Geostrophic current Global Ocean Data Analysis Project Gulf Stream Humboldt Current Hydrothermal circulation Langmuir circulation Longshore drift Loop Current Modular Ocean Model Ocean current Ocean dynamical thermostat Ocean dynamics Ocean gyre Overflow Princeton Ocean Model Rip current Subsurface ocean current Sverdrup balance Thermohaline circulation shutdown Upwelling Whirlpool Wind generated current World Ocean Circulation Experiment
Tides	Amphidromic point Earth tide Head of tide Internal tide Lunitidal interval Perigean spring tide Rip tide Rule of twelfths Slack tide Theory of tides Tidal bore Tidal force Tidal power Tidal race Tidal range Tidal resonance Tide gauge Tideline
Landforms	Abyssal fan Abyssal plain Atoll Bathymetric chart Carbonate platform Coastal geography Cold seep Continental margin Continental rise Continental shelf Contourite Guyot Hydrography Knoll Ocean bank Oceanic basin Oceanic plateau Oceanic trench Passive margin Seabed Seamount Submarine canyon Submarine volcano
Plate tectonics	Convergent boundary Divergent boundary Fracture zone Hydrothermal vent Marine geology Mid-ocean ridge Mohorovičić discontinuity Oceanic crust Outer trench swell Ridge push Seafloor spreading Slab pull Slab suction Slab window Subduction Transform fault Vine–Matthews–Morley hypothesis Volcanic arc
Ocean zones	Benthic Deep ocean water Deep sea Littoral Mesopelagic Oceanic Pelagic Photic Surf Swash
Sea level	Deep-ocean Assessment and Reporting of Tsunamis Global Sea Level Observing System North West Shelf Operational Oceanographic System Sea-level curve Sea level drop Sea level rise World Geodetic System
Acoustics	Deep scattering layer Ocean acoustic tomography Sofar bomb SOFAR channel Underwater acoustics
Satellites	Jason-1 OSTM/Jason-2 Jason-3
Related	Acidification Argo Benthic lander Color of water DSV Alvin Marginal sea Marine energy Marine pollution Mooring National Oceanographic Data Center Ocean Explorations Observations Reanalysis Ocean surface topography Ocean temperature Ocean thermal energy conversion Oceanography Outline of oceanography Pelagic sediment Sea surface microlayer Sea surface temperature Seawater Science On a Sphere Stratification Thermocline Underwater glider Water column World Ocean Atlas
Category Commons Oceans portal

• Fluid dynamics
• Oceanographical terminology