Lecture IV of VI (Claudio Piani) 3D N-S equations: stratification and compressibility, Boussinesq approximation, geostrophic balance, thermal wind.

Stratification: the third dimension Let’s start by adding a third dimension to the 2D N-S equations and allowing  to vary (they are still inviscid hence we still have no stress term): We have already met the momentum equations, note that now we express them in terms of pressure not in terms of the free surface height. We have also already met the continuity equations albeit in 1D. I am certain you are all dying to derive it by yourselves. This is the new guy…. The vertical momentum equation. It looks like the other two except it has no Coriolis term and has and added gravity term… Things will get quite a bit more complicated because this set of equations is not complete… We need a relationship between pressure and density… This is the equation of state which you all know… And since we ‘re on a role, I’ll point out that this introduces another variable, that is temperature. To deal with that we introduce yet another equation: the first principal of thermodynamics….

Boussinesq Approximation As you may have already figured out, we can understand a lot about GFD by looking at simplified versions of the NSE for which we have analytical solutions (geostrophic adjustment, gravity waves and others we are yet to see). What we intend to do today is figure out how we can simplify the 3D NSE we have just written. One of the classic simplifications is the so called “Boussinesq approximation”. In simple terms this is tantamount to assuming that the density is invariant for GFD motions in all the NSE except for in the vertical momentum equations. Naturally all simplifications come at a cost in the form of loss of accuracy. One of the side effects of adopting the Boussinesq approximation is, as we will see, that the NSE no longer can describe sound waves. In the next sections we will apply the Boussinesq approximation to all the components of the NSE.

Boussinesq Approximation Let’s start by taking a look at the continuity equation. We know from observations that density varies very little about it’s mean hydrostatic value in GFD motions. BIG SMALL We can expand the continuity equation like so:

Boussinesq Approximation The x&y components of the momentum equations are easy to deal with. All we will be doing here is to assume that  can be substituted with  0.

Boussinesq Approximation Next up is the z-momentum equation. We start in the usual way: expand and rewrite…. As for the x&y components of the momentum equations, the effects of the Boussinesq approximation on the equation of state and first principal of thermodynamics are trivial. All we will be doing is to assume that  can be substituted with  0.

Scale analysis: As they stand, these versions of the NSE are still too complex to resolve analytically. Scale analysis is a powerful tool that allows us to evaluate the size of the different terms in the equations and,hence, make further simplifications. For synoptic scale circulation we have: U=10ms -1 and L=10 6 m This referred to as geostrophic balance and, as you are sure to find out for yourselves, it is an excellent approximation for synoptic scale motion in the middle levels of the troposphere.

Geostrophic wind The values of u and v that satisfy the geostrophic approximation are referred to as geostrophic wind: Note: there is nothing particularly stratified about the geostrophic wind equations. We could have just as well derived the geostrophic approximation starting from the shallow water equations in 2D. However what we will do next with the geostrophic wind equations is to derive a relationship between the vertical rate of change of geostrophic wind and the horizontal temperature gradient. This relationship is referred to as the ‘thermal wind equation’ and it identifies a powerful constrain for GFD motions.

Thermal wind As anticipated we start by taking the vertical derivative of the geostrophic wind equations. SMALL

Thermal wind If we define the 2D geostrophic wind vector V g =(u g,v g ), then we can write the thermal wind equation in vector form: Where ‘n’ is the vertical unit vector and is the horizontal gradient. Let us assume that the temperature gradient is constant in a layer of fluid, then: The term on the LHS is referred to as thermal wind and it is the change in the geostrophic wind between two fixed levels.

Thermal wind Let’s see what the thermal wind equation can tell us about the movement of air masses: This simple diagram tells us that, when the geostrophic winds turn counter clockwise with height, there is cold advection, that is, the wind is blowing from relatively cold regions to relatively warm ones. You can work out what happens when the geostrophic winds turn clockwise with height by yourselves….

Pressure coordinates So far we have always taken for granted that the vertical coordinate should be height from the surface (z). In fact many classes of GFD motions are better described using pressure as the vertical coordinate. Also most upper level whether maps will be in pressure coordinates not height. In pressure coordinates the partial derivatives in x and y are defined along constant pressure surfaces (but you already knew that…). Let’s see what the NSE look like in pressure coordinates starting with the pressure term in the RHS of the horizontal momentum equations:

Pressure coordinates And the x&y components of the momentum equations are: Did you notice we have lost the density term ? How does that make you feel? The geostrophic wind is given by:

Pressure coordinates And the thermal wind equations? And by moving pressure to the LHS we get an even simpler form, and this time with no approximations along the way!!!!

Exercise: 1.Temperature increases southward by 10 deg. every 1000 km. Estimate the wind speed and direction at 700mb. (Hint: assume the wind is zero at 1000mb. And in geostrophic balance everywhere.) 2.There is a 10 m/s southerly wind at 850 mb and a 40 m/s easterly wind at 500 mb. Calculate the mean temperature advection in the mb layer. 3.Consider a synoptic scale low pressure system with a 1000 km radius and winds of 30 ms. Perform a scale analysis like that on slide 7. State what the order of magnitude of the error is when the geostrophic approximation is taken. Estimate the height displacement at the center of the perturbation (Hint: use isobaric coordinates)