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N. Kämpfer Atmospheric Dynamics Atmospheric Dynamics Introduction Momentum equation N. Kämpfer Geostrophic wind Thermal wind Institute of Applied Physics University of Bern Primitive equations Vorticity Outline Introduction N. Kämpfer Atmospheric Dynamics Momentum equation Introduction Momentum equation Geostrophic wind Geostrophic wind Thermal wind Thermal wind Primitive equations Vorticity Primitive equations Vorticity Introduction The circulation of the atmosphere of a planet is a key component of its climate. In case of ♁ for example: I I I Atmospheric motions carry heat from the tropics to the pole winds transport humidity from the oceans to the land Distribution of most chemical species in the atmosphere is the result of transport processes I I I I O3 is mainly produced in equatorial regions but is transported polewards H2 O enters the middle atmosphere in the tropics and is moved toward the poles CFCs are released in industrial areas and generate ozone-holes in polar regions Aerosols are produced in industrial areas and affect regions far away N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity Introduction Air motions are strongly constrained by: I I Density stratification → gravitational force resists vertical displacement Earth rotation → Coriolis force is a barrier against meridional displacements N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Transport occurs at a variety of spatial and temporal scales. In case of ♁: I global I synoptic ≈ 1000km → e.g. high and low pressure systems I mesoscale ≈ 10 - 1000km → e.g.fronts I small scale → e.g. planetary boundary layer Thermal wind Primitive equations Vorticity Introduction In order to describe the dynamical behavior of the atmosphere, we treat it as a fluid →Fundamental equations of fluid mechanics must be used N. Kämpfer Atmospheric Dynamics Introduction Circulation of a planets atmosphere is governed by three basic principles: Momentum equation Geostrophic wind 1. Newton’s law of motion Thermal wind 2. Conservation of enetgy → 1.st law of thermodynamics Primitive equations 3. Conservation of mass → equation of continuity Vorticity plus the equation of state In using these laws we must consider that we are operating in a rotating frame of reference! → we have to consider centrifugal and the Coriolis force Coordinate system It is practical to use a spherical coordinate system to describe dynamical aspects The coordinate system is rotating together with the planet We use: λ φ z R longitude in easterly direction latitude in northerly direction geometric altitude radius of planet N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations We define: r =R +z distance from the center of the planet u = r cos φdλ/dt zonal component of wind velocity ~v measured towards the east v = rdφ/dt meridional component of wind velocity ~v measured towards the north w = dz/dt vertical component of ~v Vorticity Real forces Real forces entering in the equation of motion are: 1. Pressure gradient force, F~p Whenever there is a gradient in pressure the resulting force per mass is given by 1~ F~p = − ∇p ρ F~p is normal to isobars and is directed from higher to lower pressure Distance of isobars→ measure for the pressure gradient Typical values for pressure gradient for ♁: → ca. 1mb per 8m in vertical direction and 1mb per 10km in horizontal direction 2. Frictional force, F~R is nearly proportional to wind speed N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity F~R = −a~v important in planetary boundary layer 3. Gravity force, F~G Apparent forces An object moving with velocity ~v in a plane perpendicular to the axis of rotation experiences an apparent force, called Coriolis force, F~C . F~C = 2~v × ω ~ The angular velocity is for ♁: ω = Ω = 7.3 · 10−5 sec −1 The Coriolis force per mass for zonal velocity, u, and geographic latitude ϕ is dv = −2Ωu sin ϕ dt and analogue for meridional direction du = 2Ωv sin ϕ dt where f is called Coriolisparameter f = 2Ω sin ϕ f is equal to zero at the equator and a maximum at the poles f = ±1.46 · 10−4 sec −1 N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity Apparent forces The Coriolis force in horizontal direction can be written as F~CH = −f ~k × ~v ~k is a unit vector in z-direction N. Kämpfer Atmospheric Dynamics Introduction The Coriolis force is acting normal to the direction of motion → to the right in the northern hemisphere → to the left in the southern hemisphere Momentum equation I As the force is acting normal to the direction of motion no work is performed Vorticity I The force vanishes at the equator and is maximum at the poles I Geostrophic wind Thermal wind Primitive equations The centrifugal force is another force showing up in a rotating reference frame → its effect is ”absorbed” in the gravitational force Equation of motion Considering all forces: general equation of motion per mass d~v ~p + F ~c + F ~R + F ~G =F dt N. Kämpfer Atmospheric Dynamics resp. d~v 1~ ~ × ~v − a~v + ~g = − ∇p − 2Ω dt ρ Care has to be taken what is meant by the time derivative! Introduction In the Eulerian frame this is for a fixed grid of points, actually identical to ∂~v /∂t Alternative: follow the flow In the Lagrangeian frame. This leads to the so called material derivative d ∂ ~ = + ~v · ∇ dt ∂t and therefore d~v ∂~v ~ = + ~v · ∇ ~v dt ∂t → nonlinear in ~v → difficult to forecast atmospheric state Thermal wind Momentum equation Geostrophic wind Primitive equations Vorticity Geostrophic wind Consider equation of motion in horizontal direction d~v dt N. Kämpfer ~ p + F~c + F~R = F H Atmospheric Dynamics 1~ = − ∇p − f ~k × ~v − a~v ρ Above approx. 1 km friction can be neglected Air parcel starts to move due to pressure gradient → evokes Coriolis force → deviation of track to the right → equilibrium Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity 1~ f ~k × ~v = − ∇p ρ Vector multiplication with ~k on the left allows to solve for ~v Velocity v~g is called geostrophic wind v~g = 1 ~ ~ k × ∇p ρ·f Geostrophic wind v~g = 1 ~ ~ k × ∇p ρ·f fv = 1 ∂p ρ ∂x where f = 2Ω sin ϕ − fu = 1 ∂p ρ ∂y Note: N. Kämpfer Atmospheric Dynamics Introduction Momentum equation I Geostrophic wind is parallel to isobars Geostrophic wind I On northern hemisphere → low pressure system on the left Thermal wind Primitive equations Vorticity I Wind around low pressure system in the same direction as Earth rotation I The denser the isobars the higher the wind speed I As geostrophic wind is parallel to isobars (normal to gradient) → pressure imbalance can not be changed I When friction is present → subgeostrophic wind → pressure imbalance can be changed Thermal wind From geostrophic wind equation together with ideal gas law N. Kämpfer 1 ∂p RT ∂p ∂ ln p fv = = = RT ρ ∂x p ∂x ∂x Atmospheric Dynamics Introduction From hydrostatic balance − Momentum equation ∂ ln p g = RT ∂z Geostrophic wind Thermal wind Primitive equations Neglecting vertical variations in T we get f g ∂T ∂v ≈ ∂z T ∂x f Vorticity ∂u g ∂T ≈− ∂z T ∂y These are the thermal wind equations They give relations between horizontal temperature gradients and vertical gradients of the horizontal wind when both geostrophic and hydrostatic balance apply Thermal wind N. Kämpfer Atmospheric Dynamics Introduction Geostrophic wind il-r oa< Momentum equation tl (r5 \\ Primitive equations NJ Vorticity \\ trv l I hr vv Thermal wind from Dutton: Dynamics of atmospheric motion Latitudinal temperature gradient causes an increase in the latitudinal pressure gradient → geostrophic wind speed also increases with height Thermal wind N. Kämpfer Example: The zonally average zonal wind in the lower and middle atmosphere of the Earth is close to a thermal wind Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity from Brasseur: Aeronomy of middle atmosphere from Brasseur: Aeronomy of middle atmosphere Primitive equations The whole set of equations used to describe atmospheric dynamics is called primitive equations d~v 1~ ~ × ~v − a~v + ~g = − ∇p − 2Ω dt ρ N. Kämpfer Atmospheric Dynamics Introduction Momentum equation the gas law p = ρRT Geostrophic wind Thermal wind the first law of thermodynamics Primitive equations Vorticity R T dp Q̇ dT = + dt cp ρ dt cp and the continuity equation dp ~ · ~v = 0 + ρ∇ dlt Primitive equations The whole set in spherical coordinates du uv 1 ∂φ = tan ϕ + 2Ω sin ϕv − + Fλ dt R R cos ϕ ∂λ dv u2 1 ∂φ = − tan ϕ − 2Ω sin ϕu − + Fϕ dt R R ∂ϕ ∂φ RT = ∂z H dT RT Q̇ +w = dt cp H cp N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity 1 ∂u 1 ∂(v cos ϕ) 1 ∂ρ · w + + =0 R cos ϕ ∂λ R cos ϕ ∂ϕ ρ ∂z d ∂ u ∂ v ∂ ∂ = + + +w dt ∂t R cos ϕ ∂λ R ∂ϕ ∂z where Φ = gz is the geopotential and F are friction force components Vorticity In addition to the primitive equations also equations describing vorticty in a fluid field are of importance Vorticity, ζ, in a horizontal flow is the vertical component of the rotation of the velocity field ~ z × ~v = ∂vy − ∂vx = ∂v − ∂u ζ=∇ ∂x ∂y ∂x ∂y N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Circulation is defined as Thermal wind I Z= ~ ~v · ds According Stokes: ~ n × ~v = d ∇ dA I Vorticity shows up in two cases: I I in flows that are bending in straight motion with shear ~ ~v · ds Primitive equations Vorticity Vorticity Simplest case: v = ω · r → Circulation is then I ~ = v · 2πr = 2πr 2 ω Z = ~v · ds and vorticity by division of the area element N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Z ζ = 2 =2·ω r π Geostrophic wind In this case vorticity is just two times the angular velocity In general dZ v (r ) ∂v ζ= = + dA r ∂r So far we have discussed the local vorticity, ζ. In an inertial reference system we have to add the part stemming from the planetary rotation, i.e. ω = 2Ω sin ϕ = f The absolute vorticity, η,is thus Thermal wind Primitive equations Vorticity η =ζ +f Vorticity For a geostrophic wind, we have ~vg = 1~ ~ k × ∇p ρf N. Kämpfer Atmospheric Dynamics and therefore Introduction ~ z × ~vg + f = ∇ ~z × η =ζ +f =∇ 1~ ~ k × ∇p + f ρf Momentum equation Geostrophic wind Thermal wind η= 1 2 ∇ p+f ρf It can be shown that total vorticity is concerved dη ~ H · ~v = 0 + η∇ dt where ∂η dη ~ Hη = + ~v · ∇ dt ∂t Primitive equations Vorticity Vorticity Air parcels deflected across latitudinal circles adjust to changes in local rotation → preserve absolute angular momentum as seen in an inertial frame A measure of this is potential vorticity defined as f +ζ PV = h N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind To leading order, absolute vorticity η = ζ + f is constant: the relative vorticity ζ simply being exchanged with the planetary vorticity f → Planetary waves set up Primitive equations Vorticity Hurricane on Venus N. Kämpfer Atmospheric Dynamics Introduction Momentum equation Geostrophic wind Thermal wind Primitive equations Vorticity