THE EKMAN SPIRAL

• Let's use the first-order closure ideas that we just studied to solve the equations for boundary layer motion.
• We will need to make many assumptions:
• the flow is in steady state, i.e,
• the boundary layer is horizontally homogeneous, i.e., and
• the boundary layer is statically neutral, i.e.,
• the boundary layer is barotropic, i.e., Ug and Vg are constant with height
• there is no subsidence, i.e., W = 0

after making these assumptions, the equations of motion become:

(1)

• How can we solve equation set (1)?
• First, let's define the magnitude of the geostrophic wind, G by G = [U2g + V2g]0.5
• Let's also assume that the geostrophic wind is parallel to the X axis, thus:
• G = Ug
• V2g = 0
• Let's also use first-order local closure K-theory, assuming a constant Km to eliminate the flux terms.
• Hence:    (2)
• Substituting (2) into (1) gives:

(3)

• Here we have a set of two partial differential equations...., how to solve?????
• Well, we need to specify the boundary conditions:
• U = 0 at z = 0
• V = 0 at z = 0
• U  --> G as z --> large (get above the boundary layer)
• V --> 0 as z --> large (the geostrophic flow above the boundary layer is parallel to the X axis)
• The solution to (3) using the above boundary conditions is:

(4)

where

• The velocity vectors for this solution are shown to the right.
• Notice that the velocity vectors trace out a spiral shape with height...., this is called the Ekman Spiral
• Q:  What is a physical interpretation of (4), graphically shown in the figure to the right? ANSWER

• For flow around a low, the winds converge radially into the low as a result.  This inflow into the low induces upward vertical motion.
• This process for inducing vertical motion is called Ekman Pumping.