The primitive equations are a set of nonlinear differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s that are used to approximate global atmospheric flow

Atmosphere

An atmosphere is a layer of gases that may surround a material body of sufficient mass, and that is held in place by the gravity of the body. An atmosphere may be retained for a longer duration, if the gravity is high and the atmosphere's temperature is low...

 and are used in most atmospheric model

Global climate model

A General Circulation Model is a mathematical model of the general circulation of a planetary atmosphere or ocean and based on the Navier–Stokes equations on a rotating sphere with thermodynamic terms for various energy sources . These equations are the basis for complex computer programs commonly...

s. They consist of three main sets of equations:

1. 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
2. A thermal energy equation
   Conservation of energy
   The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...
   
   : Relating the overall temperature of the system to heat sources and sinks
3. A continuity equation
   Continuity equation
   A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
   
   : Representing the conservation of mass.

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.

• is the zonal
  Zonal
  Zonal can refer to:* Zonal and meridional, directions on a globe* Zonal and poloidal, directions in a toroidal magnetically confined plasma* Zonal polynomial, a symmetric multivariate polynomial...
  
   velocity (velocity in the east/west direction tangent to the sphere)
• is the meridional velocity (velocity in the north/south direction tangent to the sphere)
• ω is the vertical velocity in isobaric coordinates
• is the temperature
  Temperature
  Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
  
  
• Φ is the geopotential
  Geopotential
  Geopotential is the potential of the Earth's gravity field. For convenience it is often defined as minus the potential energy per unit mass, so that the gravity vector is obtained as the gradient of this potential, without the minus....
  
  
• is the term corresponding to the Coriolis force, and is equal to , where is the angular rotation rate of the Earth ( radians per sideral hour), and is the latitude
• is the gas constant
  Gas constant
  The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
  
  
• is the pressure
  Pressure
  Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...
  
  
• is the specific heat on a constant pressure surface
• is the heat
  Heat
  In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...
  
   flow per unit time per unit mass
• is the precipitable water
• Π is the Exner function
  Exner function
  The Exner function is an important parameter in atmospheric modeling. The Exner function can be viewed as non-dimensionalized pressure and can be defined as:where p_0 is the pressure at the surface, usually taken as 1000 hPa; R_d is the gas constant for dry air; and c_p is the heat capacity of dry...
  
  
• is the potential temperature

Force

Force

In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...

s that cause atmospheric motion include the pressure gradient

Pressure gradient

In atmospheric sciences , the pressure gradient is a physical quantity that describes in which direction and at what rate the pressure changes the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pressure per unit length...

 force, gravity, and viscous friction

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

.