1 Governing Equations I: reference equations at the scale of the continuum by Sylvie Malardel (room 10a; ext. 2414)after Nils Wedi (room 007; ext. 2657)Recommended reading:An Introduction to Dynamic Meteorology, Holton (1992)An Introduction to Fluid Dynamics, Batchelor (1967)Atmosphere-Ocean Dynamics, Gill (1982)Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Durran (1999)Illustrations from “Fondamentaux de Meteorologie”, Malardel (Cepadues Ed., 2005)Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz
2 Overview Introduction : from molecules to “continuum” Eulerian vs. Lagrangian derivativesPerfect gas lawContinuity equationMomentum equation (in a rotating reference frame)Thermodynamic equationSpherical coordinates“Averaged” equations in numerical weather prediction models(Note : focus on “dry” (no water phase change) equations)
3 From molecules to continuum Mean free pathNewton’s second lawNavier-Stokes equationsBoltzmann equationsnumber of particleskinematic viscosity~1.x10-6 m2s-1, water~1.5x10-5 m2s-1, airEuler equationsindividual particlesstatistical distributioncontinuumFrom molecules to continuumMean value of the parameterMolecularfluctuationsContinuous variationsMean value at the scale of the continuumNote: Simplified view !
4 Perfect gas law The pressure force on any surface element containing M The temperature is defined asThe link between the pressure, the temperature and the density of molecules in a perfect gas :
6 Fundamental physical principles Conservation of massConservation of momentumConservation of energyConsider budgets of these quantities for a control volume(a) Control volume fixed relative to coordinate axes=> Eulerian viewpoint(b) Control volume moves with the fluid and always contains the same particles=> Lagrangian viewpointConservation in Eulerian depends on fluxes through the control volumeConservation assured since always the same particles contained in control volume
7 Eulerian versus Lagrangian Conservation in Eulerian depends on fluxes through the control volumeConservation assured since always the same particles contained in control volumeLagrangian : evolution of a quantity following the particules in their motionEulerian : Evolution of a quantity inside a fixed box
8 Eulerian vs. Lagrangian derivatives Particle at temperature T at position at time moves to in time .Temperature change given by Taylor series:i.e.,Letthenlocal rate of changeis the rate of changefollowing the motion.is the rate of changeat a fixed point.advectiontotal derivative
10 Mass conservation Mass flux Mass flux on left face on right face Inflow at left face is Outflow at right face isDifference between inflow and outflow is per unit volume.Similarly for y- and z-directions.Thus net rate of inflow/outflow per unit volume is= rate of increase in mass per unit volume= rate of change of density=> Continuity equation (Eulerian point of view)Mass fluxon left faceMass fluxon right faceEulerian budget :
11 Mass conservation (continued) Continuity equation, mathematical transformationContinuity equation : Lagrangian point of viewBy definition, mass is conserved in a Lagrangian volume:
12 Momentum equation : frames Newton’s Second Law in absolute frame of reference:N.B. use D/Dt to distinguish the total derivative in the absolute frame of reference.(1)We want to express this in a reference frame which rotates with the earthRotating frame“fixed” star
13 Momentum equation : velocities = angular velocity of earth.= position vector relative to earth’s centreFor any vectorEvolution in absolute frameorEvolution in rotating frame
14 Momentum equation : accelerations Then from (1):In practice:
15 Momentum equation : Forces Forces pressure gradient, gravitation, and frictionWhere = specific volume (= ), = pressure,= sum of gravitational and centrifugal force,= molecular friction,= vertical unit vectorMagnitude of varies by ~0.5% from pole to equator and by ~3% with altitude(up to 100km).
16 Spherical polar coordinates (in the same rotating frame) : = longitude, = latitude, = radial distanceOrthogonal unit vectors: eastwards, northwards, upwards.As we move around on the earth, the orientation of the coordinate system changes:Extra terms due tothe spherical curvature
18 The “shallow atmosphere” approximation “Shallowness approximation” –For consistency with the energy conservation and angular momentum conservation, some terms have to be neglected in the full momentum equation. This approximation is then valid only if these terms are negligible.with
19 Energy conservation : Thermodynamic equation (total energy conservation – macroscopic kinetic energy equation)First Law of Thermodynamics:where I = internal energy,Q = rate of heat exchange with the surroundingsW = work done by gas on its surroundings by compression/expansion.For a perfect gas, ( = specific heat at constant volume),
20 Thermodynamic equation (continued) Alternative forms:Eq. of state:R=gas constant and
21 Thermodynamic equation (continued) Alternative forms:whereis the potential temperatureThe potential temperature is conserved in a Lagrangian “dry” and adiabatic motion
22 The scale of the grid is much bigger than the scale of the continuum Where do we go from here ?So far : We derived a set of evolution equations based on 3 basic conservation principles valid at the scale of the continuum : continuity equation, momentum equation and thermodynamic equation.What do we want to (re-)solve in models based on these equations?The scale of the grid is much bigger than the scale of the continuumresolved scale (?)grid scale
23 Observed spectra of motions in the atmosphere Spectral slope near k-3 for wavelengths >500km. Near k-5/3 for shorter wavelengths.Possible difference in larger-scale dynamics.No spectral gap!Possibly investigate the Fourier representation of the heavy side function, example that single point can transferEnergy to all scales
24 Scales of atmospheric phenomena Practical averaging scales do not correspond to a physical scale separation.If equations are averaged, there may be strong interactions between resolved and unresolved scales.
25 “Averaged” equations : from the scale of the continuum to the mean grid size scale The equations as used in an operational NWP model represent the evolution of a space-time average of the true solution.The equations become empirical once averaged, we cannot claim we are solving the fundamental equations.Possibly we do not have to use the full form of the exact equations to represent an averaged flow, e.g. hydrostatic approximation OK for large enough averaging scales in the horizontal.
26 “Averaged” equationsThe sub-grid model represents the effect of the unresolved scales on the averaged flow expressed in terms of the input data which represents an averaged state.The mean effects of the subgrid scales has to be parametrised.The average of the exact solution may *not* look like what we expect, e.g. since vertical motions over land may contain averages of very large local values.The averaging scale does not correspond to a subset of observed phenomena, e.g. gravity waves are partly included at TL799, but will not be properly represented.
27 Demonstration of the averaging effect Use high resolution (10km in horizontal) simulation of flow over ScandinaviaAverage the results to a scale of 80kmCompare with solution of model with 40km and 80km resolutionThe hope is that, allowing for numerical errors, the solution will be accurate on a scale of 80kmCompare low-level flows and vertical velocity cross- section, reasonable agreementCullen et al. (2000) and references therein
28 High resolution numerical solution Test problem is a flow at 10 ms-1 impinging on Scandinavian orographyResolution 10km, 91 levels, level spacing 300mNo turbulence model or viscosity, free-slip lower boundarySemi-Lagrangian, semi-implicit integration scheme with 5 minute timestep
32 x-z vertical velocity40km resolution10km resolution averaged to 80km
33 ConclusionAveraged high resolution contains more information than lower resolution runs.The better ratio of comparison was found approximately asdx (averaged high resol) ~ ·dx (lower resol) with ~1.5-2The idealised integrations suggest that the predictions represent the averaged state well (despite hydrostatic assumption for example).The real solution is much more localised and more intense.Piotr mentions sqrt(3), theory unknown.