Chapter 2The governing systemsof equations
2.1 - The basic equations2.2 - Reynolds’ equations2.3 - Approximations to the equations‣ Hydrostatic‣ Boussinesq and anelastic‣ Shallow fluid
The equations thatare the basis for models
Dependent variables, etc.• u – east-west wind velocity• v – north-south wind velocity• w – vertical wind velocity• T – temperature• p – pressure• q v– specific humidity of water vapor• p – density• γ – lapse rate• φ – latitude• Ω – rotational frequency of Earth
Space coordinates (axes)upWx coordz coordNES
Independent variables• t – time• x• y• z
Note - All equations havetime derivatives on the left+ other equations for cloud water, rain, snow…
Other processes that are treated withseparate complex sets of equationsTurbulenceSurface heating, phasechanges, radiationCloud processes, surfaceevaporation
These basic equations• Are common to many different types ofatmospheric models- operational weather prediction models- global climate models- building-scale urban (CFD) models- research atmospheric models- models of flow over an airfoil
Variations in the equations• Almost every model uses a slightly different set ofequations.• Why?– Application to different parts of the world– Focus on different atmospheric processes – differentformulations of physics– Application to different time and spatial scales– Tailoring to customer needs– Preferences for different forms of the equations – e.g.,independent variables– Model efficiency and speed
Reynolds’ equations• The above equations apply to all scales of motion,even waves and turbulence that are too small to berepresented by models designed to simulate weatherprocesses.• Equations must be revised so that they only apply tothe larger-scale non-turbulent motions that can beresolved.• Split all the dependent variables into mean andturbulent parts, or into what is represented on amodel grid and what is notFor example
Look at the advection equation:Because we want the equation to apply tothe mean motion, apply an averagingoperator to all the terms.Apply Reynolds’ postulates:
Intuitive idea for how covariance term works.___Cool airw’T’ ?w’ > 0T’ > 0T’ < 0w’ < 0Warm air
Intuitive idea for how covariance term works.___Cool airw’T’ ?w’ > 0T’ > 0Warm airT’ < 0w’ < 0So the mean of w’T’ averagedover many parcels is positive –this represents a positive heatflux that results fromturbulence
Obtaining“Covariance” termRewrite Eq. 2.1 without Earth-curvatureterms and with only the dominant Coriolisterm, and replace Frx with something lessgeneric.
Typical way of representing Frx• τzx is the force/area, or stress, exerted in the xdirection by the fluid on one side of a z plane on thefluid on the other side.• In an hypothetically inviscid fluid there would be nocommunication between layers of fluid, and no stress• The source of this stress: molecular diffusion/mixing.
Fast-moving air(molecules)Z planeSlow-moving air(molecules)
Typical way of representing Frx• τzx is the force/area, or stress, exerted in the xdirection by the fluid on one side of a z plane on thefluid on the other side.• In an hypothetically inviscid fluid there would be nocommunication between layers of fluid, and no stress• The source of this stress: molecular diffusion/mixing.• Why the derivative of stress?WindspeedStress = F/area
A typical expression for the molecular stress isWhere µ is the dynamic viscosity coefficient.This is called Newtonian friction. If there is noshear in the fluid, viscosity produces no stress,or force per unit area, of one layer on theother. Substitute above expression for stressinto original equation to get:
Apply the averaging process to all the terms in theequation…we have only done advection so farThen, using assumptionRemember that the primes represent theturbulent motion that is superimposed onthe mean – these terms represent the effectsof the turbulence on the mean du/dt.See the parallel between themolecular stress terms and the termsthat represents the effects of theturbulence on the mean motion.
By analogy with the molecular viscosityrelatedstresses, we define turbulentstresses as follows:
Substituting these definitions of the turbulencestress into the previous equation, gives:This is the same as the original equation, exceptfor the mean-value symbols and the stress terms.We do not normally write the averaging symbols,and we neglect the molecular viscosity terms forair because they are small compared to the eddyviscosityterms. How we represent theturbulence is the subject of turbulenceparameterizations.
Approximations• Hydrostatic approximation – Constrainsvertical accelerations, ties the pressure field tothe density – this excludes sound waves,which can cause the model to run slower –applicable for only synoptic and global scales.• Boussinesq and anelastic approximations –also directly filter sound waves by couplingpressure and density perturbations, but canbe used for high-resolution models.
Shallow-fluid (shallow-water) model• Used for– Instructional purposes (e.g., for the labassignments in this course)– Testing new numerical methods for solvingthe equations.
Begin withWe’ll convert this set of equations that applies tothe atmosphere, to a set that pertains to a fluidlike water in a tank – the shallow-waterequations.
Assume incompressibility. What does this meanin terms of the continuity equation?
Now assume incompressibility. What does thismean in terms of the continuity equation?or
Now assume incompressibility. What does thismean in terms of the continuity equation?orAnother way of writing incompressibility:Now the hydrostatic equation can be rewritten:
Because the PG generates the wind, which inturn causes the Coriolis force – all forces arethus invariant with heightNow differentiate this new hydrostaticequation with respect to x, and recognize thatthe right side is a constant.This means that there is no horizontal variationof the vertical PG or vertical variation of thehorizontal PG - this is the definition ofbarotropy!
Integrate the incompressible hydrostaticequation over the depth of the fluid:where h is the depth of the fluid.
If PT = 0 or PT
We can also develop a shallow-fluidform of the continuity equation.For a horizontallower boundary
The two-dimensional shallow fluidequation set is thus:
But, for many purposes (e.g., instruction) we don’tneed/want two horizontal dimensions. So we eliminatethe y variation. BUT, if we want a mean east-west fluidflow, we need to define a constant dh/dy.
Schematic of the vertical structureof a shallow-fluid model
New terms in this chapter• Primitive equations• Primitive-equation (PE) models• Shallow-fluid model• Boussinesq approximation• Anelastic approximation