Pressure and Height Gradient

Conceptually, the pressure gradient is the rate at which pressure decreases over distance.  The partial derivative notation is consistent with our discussions of its interpretation from Tuesday.   The equivalence sign (instead of the equals sign) is used because the right hand side represents a finite difference approximation of the infinitesmal derivative on the left.  (This allows us to estimate the derivative on maps).

 

 

 

in the Cartesian or rectangular coordinate system, and

 

 

 

in the natural coodinate sytem

As you will see, the horizontal wind depends strongly on the horizontal pressure gradients.  In the example used in ERTH 260, if the air parcel has dimensions of 100 km along the x-axis, the pressure gradient would be evaluated

 

There is no pressure gradient along the y-axis for this case.

As we will see in future lectures, instead of looking at level surfaces, meteorologists and oceanographers often look at constant pressure surfaces.  The chart corresponding to the 18000 foot pressure map in x, y, p coordinates, for example, is the 500 mb chart. The variation of heights on that pressure surface is comparable to the variation of pressure on a level surface.

On a level surface, isobars can be used to express the pressure gradient.  As we will see, on a constant pressure surface, “height” contours can be used to express the same mathematical concept.  “Height” refers to the elevation (usually in meters or decameters) above sealevel at which the given pressure is found. 

On height maps,  the expressions analogous to those above are:

 

 

 

where z is the height of the given pressure surface above sealevel (meteorology) and above mean ocean surface levels (oceanography).  Oceanographers commonly use these latter expressions to develop their ideas about motion in the ocean. In meteorology (and oceanography), for example, one is required to do a finite difference estimate of the height gradient normal to the flow to calculate the geostrophic flow magnitude.

 

GeostrophicWind