The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:
A continuity equation: Representing the conservation of mass. Conservation of momentum: Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere A thermal energy equation: Relating the overall temperature of the system to heat sources and sinks The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined. In general, nearly all forms of the primitive equations relate the five variables u, v, ω, T, W, and their evolution over space and time. The equations were first written down by Vilhelm Bjerknes.
Definitions
u {\displaystyle u} is the zonal velocity (velocity in the east–west direction tangent to the sphere)
v {\displaystyle v} is the meridional velocity (velocity in the north–south direction tangent to the sphere)
ω {\displaystyle \omega } is the vertical velocity in isobaric coordinates
T {\displaystyle T} is the temperature
Φ {\displaystyle \Phi } is the geopotential
f {\displaystyle f} is the term corresponding to the Coriolis force, and is equal to 2 Ω sin ( ϕ ) {\displaystyle 2\Omega \sin(\phi )} , where Ω {\displaystyle \Omega } is the angular rotation rate of the Earth ( 2 π / 24 {\displaystyle 2\pi /24} radians per sidereal hour), and ϕ {\displaystyle \phi } is the latitude
R {\displaystyle R} is the gas constant
p {\displaystyle p} is the pressure
ρ {\displaystyle \rho } is the density
c p {\displaystyle c_{p}} is the specific heat on a constant pressure surface
J {\displaystyle J} is the heat flow per unit time per unit mass
W {\displaystyle W} is the precipitable water
Π {\displaystyle \Pi } is the Exner function
θ {\displaystyle \theta } is the potential temperature
η {\displaystyle \eta } is the Absolute vorticity
Forces that cause atmospheric motion Forces that cause atmospheric motion include the pressure gradient force, gravity, and viscous friction. Together, they create the forces that accelerate our atmosphere. The pressure gradient force causes an acceleration forcing air from regions of high pressure to regions of low pressure. Mathematically, this can be written as:
f m = 1 ρ d p d x . {\displaystyle {\frac {f}{m}}={\frac {1}{\rho }}{\frac {dp}{dx}}.}
The gravitational force accelerates objects at approximately 9.8 m/s2 directly towards the center of the Earth. The force due to viscous friction can be approximated as:
f r = f a 1 ρ μ ( ∇ ⋅ ( μ ∇ v ) + ∇ ( λ ∇ ⋅ v ) ) . {\displaystyle f_{r}={f \over a}{1 \over \rho }\mu \left(\nabla \cdot (\mu \nabla v)+\nabla (\lambda \nabla \cdot v)\right).}
Using Newton's second law, these forces (referenced in the equations above as the accelerations due to these forces) may be summed to produce an equation of motion that describes this system. This equation can be written in the form:
d v d t = − ( 1 ρ ) ∇ p − g ( r r ) + f r {\displaystyle {\frac {dv}{dt}}=-({\frac {1}{\rho }})\nabla p-g({\frac {r}{r}})+f_{r}}
g = g e . {\displaystyle g=g_{e}.\,}
Therefore, to complete the system of equations and obtain 6 equations and 6 variables:
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