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., U_g and V_g 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 = [U²_g + V²_g]^0.5  
• Let's also assume that the geostrophic wind is parallel to the X axis, thus:
  • G = U_g
  • V²_g = 0
• Let's also use first-order local closure K-theory, assuming a constant K_m 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.