The simplifed momentum equations

1. The simplifed momentum equations Height coordinates Pressure coordinates

2. In this section we will examine the effect of friction on the wind in the boundary layer. We will derive the equation describing the “Ekman Spiral” and learn how friction always leads to the demise of high and low pressure systems

3. Planetary Boundary Layer The layer of the atmosphere in which the flow field is strongly influenced directly by interaction with the surface of the earth PBL is dominated by turbulent flows, which are not easily characterized (mathematically) as flows in the free atmosphere above the boundary layer

4. Boundary layer meteorologists roughly divide the PBL into three regions 1. The viscous sub-layer The layer extending from the surface to a few millimeters above the surface Characterized by: -molecular diffusion -extreme shear -wind at surface (molecular scale) = 0 2. The surface layer The layer extending from the top of the viscous sub-layer to about 10% of the depth of the PBL Characterized by: -vertical momentum transfer by turbulent eddies -not directly dependent on Coriolis and PG forces 3. The Ekman layer The layer extending from the top of the surface layer to the top of the PBL Characterized by: -turning wind with height as the effect of friction diminishes and the wind approaches its geostrophic value

5. Because of turbulence, there is not simple solution to describe planetary boundary layer flows We will limit our investigation in this class to the “Ekman Spiral” with an eye toward understanding how friction will impact large scale high and low pressure systems To learn more: Take Boundary Layer Meteorology!

6. We consider now an elegant (but unfortunately not very accurate) method of describing the wind in the boundary layer, due originally to V. Walfrid Ekman. (in fairness to Ekman, he derived this to show how currents change with depth in the ocean – the formulation worked well until meteorologists turned it upside down)

7. Horizontal Momentum Equations Let us assume that: (1) the flow is steady state (2) for simplicity that the flow above the boundary layer is west-east (3) the flow above the PBL is geostrophic (4) K is constant (actually varies with z) (5) f is constant Making use of the geostrophic relationships:

8. With these assumptions, our equations become (1) (2) We will now do a mathematical trick to solve for the Ekman spiral Lets multiply (2) by the imaginary number and add it to (1) This is a second order inhomogeneous ordinary differential equation We will solve the equation with boundary conditions:

9. Let’s simplify the look of the equation by using This is a simple second order inhomogeneous ordinary differential equation To find the general solution, we must find a single particular solutionto the equation and a complimentary solution to the corresponding homogeneous equation:

10. Let’s seek a particular solution first: Clearly, one solution of the inhomogeneous equation is obtained by assuming that N is independent of z. This reduces the equation to Now let’s seek a complementary solution to the homogeneous equation If we seek a solution of the form we get

11. Then  can have two values: Let’s define The general solution to the homogeneous equation is therefore or

12. The first term on the right, The only way the solution can be finite is for The complete solution is the sum of the particular solution to the inhomogeneous equation and the general solution to the homogeneous equation

13. We will solve the equation with boundary conditions: Setting z = 0

14. Dividing into the real and imaginary parts For simplicity, we will now assume that the geostrophic wind is zonal (west-east) so the vg = 0

15. These equations describe the Ekman Spiral Let’s look at an example and they we will try to determine what these equations are telling us about the wind in the boundary layer

16. Measurements we will look at were taken over land where the cloud streets formed

17. The geostrophic wind was parallel to the cloud streets Ekman Spiral 4 m/s 14 m/s Observed wind Each red dot progresses another 100 m above surface

18. Most important aspects of the Ekman solution • There is cross-isobar flow towards low pressure. • The velocity = 0 at the lower boundary. • The velocity tends to the geostrophic flow at top of PBL. • For small z, we have so that the flow near the surface is 45 to the left of the geostrophic flow.

19. Remarks on the Ekman Solution • The Ekman theory predicts a cross-isobar flow of 45 at the lower boundary. This is not in agreement with observations • Better agreement can be obtained by coupling the Ekman layer to a surface layer where the wind direction is unchanging and the speed varies logarithmically See: Brown, R., 1970: A Secondary Flow Model for the Planetary Boundary Layer. J. Atmos. Sci., 27, 742–757. (source of figures)

20. The geostrophic wind was parallel to the cloud streets Ekman Spiral 4 m/s 14 m/s Observed wind Theory with surface layer coupled to Ekman layer

21. Implication of Ekman theory for synoptic scale systems

22. Mass convergence into low pressure centers and mass divergence out of high pressure centers will eventually destroy the weather systems while forcing upward vertical motion in low and downward vertical motion in high