FLOW STABILITY

• Another important question in boundary layer meteorology is determining when unstable flows become or remain turbulent.
• At the same time, if you have a laminar flow, will it remain laminar, or will it become turbulent?

To answer these questions, we first need to discuss the idea of stability.

• In previous met courses, you have discussed the idea of local convective static stability:
• local lapse rates are computed and compared to the dry and moist-adiabatic values.
•  In the boundary layer, however, there are more than just one type of instability.
• In addition to the convective instability, there is also a mechanical, or shearing instability.
• These two stability mechanisms are very much related to the buoyancy and mechanical TKE production terms in the TKE budget equation.
• Lets look at each of these stability mechanisms in more detail.

Convective Stability

• You are already familiar with this concept.
• If a parcel of air is less dense than it's local environment, it will rise due to the buoyancy force.
• If a parcel of air is more dense than it's local environment, it will sink due to the buoyancy force.

Dynamic Stability

Even if the air is statically stable, inhibiting convectively-driven turbulence, a dynamical instability can occur that involves a region of wind shear in the flow.

How does it work?

1.  Initially, wind shear exists across an interface separating layers of fluid with different densities.  This flow is initially laminar.

2.  If a critical value of shear is reached, then the flow becomes dynamically unstable and gentle waves begin to form on the interface.  The crests of these waves are normal the the shear vector orientation.

3.  These waves continue to grow in amplitude, eventually reaching a point where they begin to roll up or "break".  This breaking wave is called a Kelvin-Helmholtz (KH) wave and has been produce through a Kelvin-Helmholtz shearing instability.

4.  Within each wave, there exists some lighter fluid that been rolled under denser fluid, resulting in patches of static instability.  These breaking waves are therefore, very turbulent due to both dynamic and convective instabilities.

5.  The turbulence spreads throughout the layer, causing the two fluids to mix.  The turbulence also transfers momentum that results in weaker shear across the interface.

6.  The turbulence dissipates within this layer, eventually allowing the flow to become laminar again.

This sequence of events is suspected to occur during the onset of Clear-Air Turbulence (CAT)

Q:  That said, where in the atmosphere would you expect CAT to occur?? Answer

Areas of CAT can extend for hundreds of kilometers but are usually confined to shallow layers in the vertical (tens to hundreds of meters).

That is why it is possible for a pilot to ascend or descend to quickly get out of a region of turbulent air.

KH instability can be observed visually if there is enough moisture in the air to produce billow clouds.  Although not very common, there have been a few instances when very striking billow clouds have been photographed.

Q:  Given a shear layer across an interface separating fluids with different densities, under what conditions will KH instability occur?  Under what conditions will the flow be laminar?  What sector of meteorology would be interested in knowing the answers to these questions????

Do an image google on either KH waves or billow clouds for more image examples

The Richardson Number

• Given a layer of air with some value of wind shear, dU/dz and static stability dqv/dz.
• Again, the question is under what conditions is this layer of air susceptible to KH instability and the generation of turbulence?
• Theoretical treatment of this subject has shown that the the following dimensionless number can be used to answer this question:

(1)

Ri is the gradient Richardson number, or just Richardson number.

Q:  How does one physically interpret Ri?

Theoretical and laboratory results have shown the Richardson number can be used to determine whether or not the flow will become turbulent due to the dynamic (shearing) instability.

Specifically, the dynamic stability criteria are:

• Laminar flow becomes turbulent when Ri < Rc

• Turbulent flow becomes laminar when Ri > Rt

where Rc = 0.25 and Rt = 1.0.

In practice, one would use sounding data and a finite difference form of (1) to compute Ri.  The finite difference form of (1) is:

(2)

where RB is the bulk Richardson number.  This is the form of the Richardson number used most often in Meteorology.

Notice that the above stability criteria for the gradient Richardson number are for an infinitely shallow layer of air and therefore, can not be exactly applied when calculating the bulk Richardson number.  For example, KH instability could occur in a layer of air of finite depth with RB greater that 0.25.

Questions about either convective or dynamic stabilities??

Let's now do the dynamic stability lab.