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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 pressuretendency
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
∇·V_{H} + ∂ω/∂p =0
where ω=dp/dt
and thus
∂ω/∂p =  ∇·V_{H}
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 + V_{H}·∇p + ω∂p/∂z
but, since ∂p/∂z = ρgw this becomes
dp/dt = ∂p/∂t + V_{H}·∇p  ρgw
Furthermore, the horizontal wind V_{H} may be represented as the sum
of a geostrophic component V_{g}
and an ageostrophic component V_{a}.
But the advection of pressure by the geostrophic wind is zero because V_{g}
is proportional to k×∇p and thus is perpendicular to
∇p.
Therefore,
V_{H}·∇p = V_{g}·∇p + V_{a}·∇p
= V_{a}·∇p
Thus
∂p/∂t = ω  ∇·V_{a} + 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: