Physics of the Atmosphere I

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1 Physics of the Atmosphere I WS 2008/09 Ulrich Platt Institut f. Umweltphysik R. 424 heidelberg.de

2 Last week The conservation of mass implies the continuity equation: ρ + ( ρ v) = 0 t Thermal wind occurs due to horizontal temperature gradients Direct thermal wind is a small scale process (Coriolis force can be neglected) and leads to a circulation, with wind blowing from cold to warm surfaces Geostropic thermal wind occurs on large scales (e.g., latitudinal temperature gradient) and causes a geostrophic wind pattern. The wind speed increases linearly with altitude. Universität Heidelberg Institut für Umweltphysik

3 Contents Introduction Literature - Vertical structure of the atmosphere Adiabatic processes - Vertical stability Atmospheric radiation: Absorption, scattering, emission Atmospheric radiation: The energy budget of the atmosphere Atmospheric dynamics: Navier-Stokes equation Atmospheric dynamics: Continuity equation, thermal wind Atmospheric dynamics: Vorticity Atmospheric dynamics: The planetary boundary layer Atmospheric circulation: Global circulation patterns, planetary waves Atmospheric circulation: The ENSO phenomenon Diffusion and turbulence: Molecular diffusion, basics of turbulence Diffusion and turbulence: Theorem of Taylor, correlated fluctuations Diffusion and turbulence: Diffusion of scalar tracers Near-surface dynamics: Wind profile, influence of surface friction

4 Outline for Today The concept of vorticity: Circulation The Definitions of Atmospheric Vorticity Potential Vorticity (PV) Examples Rigid Rotation Flow on curved trajectory Shear flow Observations of PV

5 The Concept of Vorticity Apart from the conservation of energy and momentum, the angular momentum is a conserved quantity in a dynamical system In atmospheric physics, the conservation of angular momentum is expressed as the conservation of the vortex strength of the wind vector field Different definitions of the vortex strength exist

6 Definition of Vorticity The curl of the wind vector field is only important in the horizontal since the vertical extent of the atmosphere is very small The relative vorticity (relative vortex strength) is defined as the z-component of the curl of the wind vector field: with: v y = z x ( v) ζ = = ( rot v) ( v) v x y z z

7 Vorticity and Circulation The curl of a vector field perpendicular to a given surface A with normal vector n is related to the circulation Z of the vector field (i.e., the closed path integral along the border S of A), Z = v ds n A via Z(A) d v = lim = vds A da ( ) n A 0 S(A) A S

8 Vorticity and Circulation Circulation in Cartesian coordinates: ( ) ( ) ( ) ( ) dz = v x y dx + v y x + dx dy v x y + dy dx v y x dy z ζ y y + dy x y x + dx x

9 Vorticity and Circulation ( ) ( ) ( ) ( ) dz = v y dx + v x + dx dy v y + dy dx v x dy x y x y v y v = ( ) ( ) + ( ) + ( ) + x y x v x y dx v y x dy v y x dx dy v x y dy dx v x v y dx dy y x = With da = dx dy the above equation becomes: ζ dz v v da x y y = = Note that these findings are a special case of the Stokes theorem: Sign of ζ: Positive for counterclockwise rotation Negative for clockwise rotation S vds = v da ( ) A x

10 Vorticity of a Rigid Rotator Velocity at distance r from the axis: v = ω Circulation: r 2 Z = vds = 2π rv = 2πr ω Thus, the vorticity is r dz Z 2πr ω ζ = = = = 2 2 da π r π r 2 2ω The vorticity of a rigid rotator is twice its angular velocity Example: High pressure system, R = 500 km, v=10 ms -1 ω = v/r 10/ = s -1 ζ = 2 ω = s -1

11 Vorticity of a curved trajectory General case of a curved trajectory Radius of curvature as fu. of ds: dϕ v(r) r r + dr Radius of curvature is always perpendicular to the velocity: dr v = 0 Circulation around infinitesimal area da = r dr dφ: dz = v(r)r dϕ + v(r + dr) (r + dr)dϕ v = v(r)r dϕ + v(r) + dr (r + dr)dϕ r v = v(r)dr dϕ + r dr dϕ + O dr r ( 2 )

12 Vorticity of a Curved Trajectory Dividing by the infinitesimal area da = r dr dφ yields: v v(r)dr dϕ + r dr dϕ dz r v(r) v ζ = = = + da rdrdϕ r r Note that in case of a rigid rotator with dv v = ω r = ω dr this yields ζ = ω + ω = 2ω which is the same result as previously inferred

13 Vorticity of shear winds We assume flow (wind) in x-direction, see figure below (analogous considerations can be made for y- direction). Thus v y = 0 and consequently also: v y = 0 x Thus the vorticity is v v v ζ = ( v) = = z x y y y x x y A straight line connecting different air parcels in y-direction will rotate due to wind shear x

14 The Absolute Vorticity The dynamics of the atmosphere is described in a rotating coordinate system The Earth is a rigid rotator with angular velocity Ω Local vorticity in z-direction at latitude φ is given by the Coriolis parameter f = 2 Ω sin φ (see previously discussed vorticity of a rigid rotator) Thus the absolute vorticity η of the wind field is the sum of the relative vorticity ζ (measured relative to the terrestrial coordinate system) and the Coriolis parameter f: η = ζ + f v + 2Ω sinϕ ( ) z

15 The Vorticity Equation To infer a continuity equation for the absolute vorticity, we start with the Navier-Stokes equation The gravitational term is zero since only horizontal motion is considered (v z = 0),. Furthermore, the external forces (pressure gradient, friction and others) are summarised as f ext : v ( v ) ρ + v = f ext + 2ρ v Ω t external forces Coriolis force The advectional term can also be written as: ( ) 1 2 v v = v v ( v) 2 Thus, using the Coriolis parameter f = 2 Ω sin φ, we have: v v 1 ( ) x v v v f v y = fext t 2 ρ

16 The Vorticity Equation Now we calculate the z-component of the curl of this equation. The different terms yield: 1. v ( v) ζ η = = = t t z t t z In the last step, we have used η = ζ + f and f/ t = 0 2. v = 0 ( 2 ) since the curl of a gradient field is zero

17 ( ) 3. The Vorticity Equation 0 v v = v 0 z ζ { } v y ζ = v x ζ 0 = x z ( vx ζ ) ( vy ζ ) x y = ζ vx ζ z y v ζ v ζ vy x x y y x ζ ( ) ( ) = Hv v Hζ with H = y 0

18 The Vorticity Equation 4. = y v y f v x f x 0 z ( vx f) ( vy f) v x f v y f = f vx f vy x x y y = f v v f ( ) ( ) H H

19 The Vorticity Equation Putting these terms together, we have: η 1 + ζ v + v ζ + f v + v f = f t ρ and finally, with η = ζ + f, η 1 + η H v + v Hη = fext t ρ ( ) ( ) ( ) ( ) ( ) H H H H ext ( ) ( ) ( ) η 1 + ( ) ( ) H η v = fext t ρ z This is a continuity equation for the absolute vorticity, known as the vorticity equation Source term for the vorticity is the curl of the external force field Note that pressure gradients do not induce vorticity since the curl of the pressure force f p = p is zero or z z

20 The Potential Vorticity The conservation of vorticity expressed by the vorticity equation is only valid if there is no vertical movement of air. The vorticity is not conserved if lifting of air (i.e., v z 0) occurs. By combining the vorticity equation with the conservation of mass a new quantity is derived: The Potential Vorticity (PV) For this quantity a more general conservation law can be derived, which is directly applicable to the atmosphere The conservation law for the potential vorticity is valid for a barotropic atmosphere, i.e. if the isolines of pressure and temperature are parallel Two equivalent definitions for the potential vorticity after Ertel and Rossby exist (with the former being more popular in atmospheric sciences)

21 Rossby s Potential Vorticity Rossby s potential vorticity is a quantity, related to an air column of finite vertical extent z = z 2 z 1 We assume that the air parcel is bounded at its bottom and top by surfaces of defined pressure or temperature. p = p(z 2 ) p(z 1 ) is the pressure difference between top and bottom of the air column Integrating the continuity equation, z 2, p 2 ρ + ( ρ v) = 0 z 2, p 2 t which expresses the conservation of mass, from z 1 to z 2 yields (after several z 1, p 1 re-arrangements, see Roedel): 1 d p z 1, p 1 H v = p dt This equation relates temporal changes in vertical pressure differences to the horizontal divergence of the wind field

22 Rossby s Potential Vorticity Now we combine the continuity equation in the form 1 d p H v = p dt with the vorticity equation dη 1 dη + η Hv = 0 or Hv = dt η dt to 1 d p 1 dη η d p 1 dη + = 0 or + = 0 p dt η dt p dt p dt ( ) 2 This is equivalent to dz dt with Rossby s potential vorticity quantity = 0 or Z const. r r = Z R η = p being a conserved

23 Ertel s Potential Vorticity Assuming the upper and lower boundary layers of the airparcel having the potental temperatures Θ 1 and Θ 2, respectively. The Difference Θ is conserved during adiabatic ascent or descent. dθ Θ = Θ 2 Θ 1 = ( z 2 z 1) = const dz Neglecting the change in density due to different actual temperatures, we can express the altitude difference as pressure difference: p z z = inserting z z g ρ ( ) in the above Eq. yields: g ρ p = const inserting this expression of p in Eq. for ZR : dθ dz η dθ η dθ ZE = = const. or Z * E = = const ' ρ dz ρ dp dp= g ρ dz

24 Some Properties of (Ertel( Ertel s) ) PV η dθ η dθ ZE = = const. or Z * E = = const ' ρ dz ρ dp dp= g ρ dz Dimension: [ Z ] Frequently: 10 E s K K m = = Kg m Kg s 3 m 2 K m = 1PVU (PV-Unit) Kg s

25 Z E Example: : Change of Wind Direction in Flows Across Obstacles η dθ = = const. ρ dz Change of wind direction in a flow across an obstacle (hill) due to the conservation of PV. Here a constant coriolis parameter is assumed. Note that the direction of deflection is independent of the direction of flow. Adapted from: W. Roedel, 1992, p. 107.

26 Adiabatic Flow Over a Mountain Range Z E η dθ = = ρ dz const. Holton (1992) uniform zonal flow initial lifting of Θ 0 +dθ layer stretching of Θ 0 +dθ layer horizontal spread of vertical displacement at top of column development of leewave due to changes in f

27 PV and Potential Temperature Gradients Stratosphere, dθ/dz 20 K/km Troposphere, dθ/dz 5 K/km Stratosphere: dθ/dz higher, density lower than troposphere

28 Comparison of Absolute and Potential Vorticity at an Altitude of ca. 8 km. ETA-32 Map for Nov 6, 2001, 10:00

29 PV in the Arctic Polar Vortex

30 PV in the Arctic Polar Vortex

31 Summary The conservation of vorticity (curl of the wind field) expresses the conservation of angular momentum The absolute vorticity (sum of relative vorticity and coriolis parameter) also considers the rotation of the Earth The vorticity follows a continuity equation (the vorticity equation) with the curl of the external force field as source term The potential vorticity (PV) is an important concept in atmospheric dynamics. It connects the continuity equation with the vorticity equation and is also valid for vertical movements PV is a conserved quantity in a barotropic atmosphere

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