Key Thematic Elements Underlying framework for NWP (Chs. 2-3) Physical process parameterizations (Chs. 4-5) Model initialization methods (Ch. 6) Applications of numerical models (Chs. 7+)
Course Structure We will make heavy use of the assigned text for this course. I anticipate lecturing for approximately 60 min each class, leaving 15 min for discussion and questions. Thus, please do read the section(s) to be covered in class ahead of time!
Basic Equation Set Horizontal Momentum Equations φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction advection terms curvature terms friction Coriolis termspres. grad. time derivs.
Basic Equation Set Vertical Momentum Equation φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction advection terms curvature term friction Coriolis termpres. grad. time deriv. gravity
Basic Equation Set Thermodynamic Equation γ = lapse rate of temperature, γ d = dry adiabatic lapse rate, Q = diabatic heating rate advection terms dry adiabatic term diabatic heating time deriv. or, alternately…
Basic Equation Set Continuity Equations (i.e., mass – top – and water vapor – bottom – are neither created nor destroyed) q v = water vapor mixing ratio, Q v = source/sink of qv due to phase changes advection termstime derivs. divergence term
If you can solve the set of equations (often referred to as the primitive equations) given in the past few slides, you can do NWP! …but… …how do we actually solve these equations???
What Do We Need? How do we represent these equations on a map? How do we integrate them in time? How do we integrate them in space? How do we handle resolvable versus unresolvable processes? How do we handle diabatic processes? How do we handle friction? How do we handle microphysical phase changes (water vapor as well as other species – cloud, ice, graupel, rain, etc.)? How do we obtain our initial atmospheric state? …all among many relevant questions we will address!
Another scary thought: Nearly everything we describe from here on out involves some sort of approximation. This is true for the equations themselves, the methods used to solve them, the initial and boundary data used to drive the model, and so on.
Prognostic vs. Diagnostic Prognostic: any equation with a time-derivative is a prognostic equation; it can be integrated in time to produce a prediction Diagnostic: any equation without a time-derivative is a diagnostic equation; it can only be used to diagnose what is happening at a given time
Vertical Coordinate As given before, our vertical coordinate is height (z) – Note: the comment on pg. 7 is wrong (should be z, not p) Other vertical coordinates may be used if appropriate substitutions are made… – Pressure (p) – Potential temperature (θ) – Terrain-following (σ) We’ll discuss these further in a later chapter.
Reynolds Averaging The equations we outlined earlier are valid on all scales of motion But, in a numerical model, we have a grid with finite horizontal and vertical resolution… This leads to there being two scales of physical processes: those that we can resolve (larger) and those that we cannot resolve (smaller)
Reynolds Averaging The goal of Reynolds averaging is to separate out the resolvable and unresolvable scales of motion. We do so by splitting our dependent variables (u, T, p, etc.) into mean (resolved) and turbulent (perturbation/unresolved) components, e.g.,
Reynolds Averaging General idea: – Substitute such expressions into the primitive equations – Take the mean of each of the primitive equations – Simplify the result using Reynolds’ postulates Reynolds’ postulates… mean of all perturbations is zero mean of a mean is equal to the mean mean operator is commutative unless both variables are turbulent/perturbation components
Reynolds Averaging: Example Equation (2.11), dropping frictional parameterization… We desire to substitute for u, v, w, p, and ρ, average the entire equation, and ultimately simplify the result using Reynolds’ postulates.
Reynolds Averaging: Example resolvable scalesaggregate (mean) effects on unresolvable scales
Reynolds Averaging: Example Typically, the unresolvable scale terms are written in terms of turbulent stresses. – See also: discussion related to equations (2.14)-(2.16). – These stresses are typically approximated utilizing physical parameterization packages, to be covered in later lectures. We commonly drop the overbars from the primitive equations, making the Reynolds average(s) implicit.
Reynolds Averaging: An Aside The discussion of Reynolds averaging in the text discusses two things that we have neglected: – Frictional parameterization – Tensor notation We will also visit these topics later in the course.
Squelching Acoustic Waves The biggest thorn in the side of modelers is acoustic waves… – Acoustic waves aren’t meteorologically relevant. – But, their very high frequency means that a very short time step (and a LOT of computer power) is needed to solve the primitive equations… – …unless we do something to filter them out exclusively. Recall: the restoring mechanism for acoustic waves is their compressibility.
Squelching Acoustic Waves In other words, the propagation of acoustic waves is reliant upon the density adjusting to horizontal compression and expansion within the waves. To filter out acoustic waves, we need to eliminate the possibility of compressibility. Specifying conditions under which density can (and cannot) vary enables us to filter out acoustic waves.
Squelching Acoustic Waves Method 2: Boussinesq Approximation – Assumes that density is constant except where it is tied to buoyancy (or gravity). – Equivalently obtained (in part) by replacing (2.5) with: – This relates density perturbations to temperature, rather than pressure, perturbations. Note: Boussinesq continuity equation conserves volume (not mass), not necessarily a good trade-off!
Squelching Acoustic Waves Method 3: Anelastic Approximation The Boussinesq approximation can be viewed as a simplified subset of the anelastic approximation. Obtained (in part) by replacing (2.5) with:
Summary We have introduced the basic equation set used for numerical weather prediction. We have briefly described what is needed in order to solve this equation set. We have outlined the need for separating the resolvable from the unresolvable scales of motion. We have described the need for filtering out acoustic waves from the primitive equations.