Thayer Watkins
Silicon Valley

The Pressure Tendency Equation

Expressions such as, "The barometer is falling" or "The barometer is holding steady" refer to ∂p/∂t, the pressure tendency. These expressions imply pressure tendency means something about changes in weather so an equation for predicting the pressure tendency could have some possible value for weather prediction.

The pressure-tendency equation is an equation giving the rate of change of pressure in terms of the cumulative divergence of the wind velocity field. It is derived directly from the continuity equation expressed in terms of pressure rather than height as the vertical coordinate.

The form of the continuity equation in such coordinates is

.VH + ∂ω/∂p =0
where ω=dp/dt
and thus
∂ω/∂p = - .VH

This equation can be integrated with respect to p downward from the top of the atmosphere where p=0 and ω=0 to pressure level P to give the result:

Since ω = dp/dt it necessary to get a correction for advection in order to determine the pressure tendency ∂p/∂t. The required equation is:

dp/dt = ∂p/∂t + VH.p + ω∂p/∂z
but, since ∂p/∂z = -ρgw this becomes
dp/dt = ∂p/∂t + VH.p - ρgw

Furthermore, the horizontal wind VH may be represented as the sum of a geostrophic component Vg and an ageostrophic component Va. But the advection of pressure by the geostrophic wind is zero because Vg is proportional to k×p and thus is perpendicular to p. Therefore,

VH.p = Vg.p + Va.p
= Va.p


∂p/∂t = ω - .Va + ρgw

The advection of the ageostrophic wind is of the second order degree of smallness compared with the pressure tendency and can be neglected. At ground surface where w=0 then the tendency for the surface pressure P is given approximately by: