GEF 2220: Dynamics J. H. LaCasce Department for Geosciences University of Oslo GEF 2220: Dynamics p.1/315
Primitive equations Momentum: t u + u u + f yw f z v = 1 ρ t v + u v + f zu = 1 ρ t w + u w f yu = 1 ρ x p + ν 2 u y p + ν 2 v z p g + ν 2 w GEF 2220: Dynamics p.2/315
Primitive equations Continuity: Ideal gas: ρ + u ρ + ρ u = 0 t Thermodynamic energy: p = ρrt dt c v dt + pdα dt = c dt p dt αdp dt = dq dt GEF 2220: Dynamics p.3/315
Primitive equations Six equations, six unknowns: (u,v,w) velocities p pressure ρ density T temperature GEF 2220: Dynamics p.4/315
Primitive equations Momentum equations F = ma Continuity ρ Thermodynamic energy equation T Ideal gas law relates ρ, p and T GEF 2220: Dynamics p.5/315
Prediction Solve the equations numerically with weather models Issues: Numerical resolution Vertical coordinate Small scale mixing Convection Clouds Goal: forecasting GEF 2220: Dynamics p.6/315
Dynamics Solve a simplified set of equations Identify dominant balances Simplify the equations Obtain solutions (analytical, numerical) Look for similiarities with observations Goal: understanding the atmosphere GEF 2220: Dynamics p.7/315
Momentum equations Take the x-momentum equation: t u = 2 t 2x = 1 ρ i F i which is like: a x = 1 m i F i GEF 2220: Dynamics p.8/315
Momentum equations Two types of forces: 1) Real 2) Apparent Two ways to write the derivative: 1) Lagrangian 2) Eulerian GEF 2220: Dynamics p.9/315
Derivatives Consider an air parcel, with temperature T T = T(x,y,z,t) The change in temperature, from the chain rule: dt = T T dt + t x dx + T y dy + T z dz GEF 2220: Dynamics p.10/315
Derivatives So: dt dt = t T + u T x + v T y + w T z = T + u T t d dt t is the Lagrangian derivative + u is the Eulerian representation GEF 2220: Dynamics p.11/315
Lagrangian T(t ) 1 T(t ) 2 GEF 2220: Dynamics p.12/315
Eulerian (x,y,z) T(t) GEF 2220: Dynamics p.13/315
Real forces Pressure gradients Gravity Friction GEF 2220: Dynamics p.14/315
Pressure gradient δy x 0 y 0 z 0 δz δx δv = δx δy δz GEF 2220: Dynamics p.15/315
Pressure gradient Use Taylor series: G(x 0 + δx) = G(x 0 ) + G x δx + 1 2 2 G x 2 δx2 +... Pressure on the right side of the box: p = p(x 0,y 0,z 0 ) + p x Pressure on left side of the box: p = p(x 0,y 0,z 0 ) p x δx 2 +... δx 2 +... GEF 2220: Dynamics p.16/315
Pressure gradient The force on the right hand side (directed inwards): On left side: So the net force is: p = [p(x 0,y 0,z 0 ) + p x p = [p(x 0,y 0,z 0 ) p x δx 2 ]δyδz δx 2 ]δyδz F x = p x δx δy δz GEF 2220: Dynamics p.17/315
Pressure gradient The volume weighs: m = ρδx δy δz So: a x = F x m = 1 ρ p x Same derivation for the y and z directions. GEF 2220: Dynamics p.18/315
Momentum equations Momentum: du dt = 1 ρ dv dt = 1 ρ dw dt = 1 ρ x p +... y p +... z p +... GEF 2220: Dynamics p.19/315
Gravity Acts downward (toward the center of the earth): a z = F z m = g dw dt = 1 ρ z p g GEF 2220: Dynamics p.20/315
Friction τ + zx δτ zx δ z δ z 2 δ z τ zx τ zx δτ zx δ z δ z 2 δ y δ x GEF 2220: Dynamics p.21/315
Friction Net viscous force (stress area) of the boundaries acting on the fluid : (τ zx + τ zx z δz 2 )δx δy (τ zx τ zx z δz 2 )δx δy = τ zx z δx δy δz Divide by the mass of the box: F zx = 1 ρ τ zx z GEF 2220: Dynamics p.22/315
Friction Similar derivations for τ zy, τ xx,... Applied to x-direction: du dt = 1 ρ ( τ xx x + τ yx y + τ zx z ) GEF 2220: Dynamics p.23/315
Friction z=l u(l) u(z) z=0 u(0)=0 GEF 2220: Dynamics p.24/315
Friction Experiments show that force required to keep plate moving at a constant velocity is: F zx u(l)a l = µ u(l)a l where µ is the viscosity coefficient GEF 2220: Dynamics p.25/315
Friction Each fluid layer exerts a stress on layer below it: u(z + δz)a Fzx = µ( δz As δz 0: u(z)a ) µa( δu δz δz ) τ zx = F zx A = µ u z GEF 2220: Dynamics p.26/315
Friction So: 1 ρ τ zx z = 1 ρ z (µ u z ) If ν = µ/ρ is constant: 1 ρ τ zx z = ν 2 z 2u GEF 2220: Dynamics p.27/315
Friction At sea level: ν = 1.46 10 5 m 2 /sec Applied to x-direction: du dt = ν 2 z 2u Friction acts to diffuse momentum, reducing the vertical shear. GEF 2220: Dynamics p.28/315
Momenutum equations With all the friction terms, have: du dt = ν ( 2 2 2 x2u + y2u + z 2u) = ν 2 u dv dt = ν ( 2 2 2 x2v + y2v + z 2v) = ν 2 v dw dt = ν ( 2 2 2 x2w + y2w + z 2w) = ν 2 w GEF 2220: Dynamics p.29/315
Friction Mixing is due to molecular motion Creates a boundary layer, near the earth s surface But the molecular layer is only a few centimeters thick Molecular mixing unimportant for weather-scale motion GEF 2220: Dynamics p.30/315
Planetary boundary layer Away from the surface, momentum is mixed by eddies Usually represent the mixing in the same way: F x = K m 2 u where K m is an eddy viscosity K m ν GEF 2220: Dynamics p.31/315
Momentum equations So far: du dt = t u + u u = 1 ρ dv dt = t v + u v = 1 ρ dw dt = t w + u w = 1 ρ x p + ν 2 u y p + ν 2 v z p g + ν 2 w GEF 2220: Dynamics p.32/315
Apparent forces Space Earth GEF 2220: Dynamics p.33/315
Rotation Ω Α δθ δα γ γ GEF 2220: Dynamics p.34/315
Rotation δθ = Ωδt Assume Ω = const. (reasonable for the earth) Change in A is δa, the arc-length: δ A = A sin(γ)δθ = Ω A sin(γ)δt = ( Ω A)δt GEF 2220: Dynamics p.35/315
Rotation So: d A dt = Ω A This is the motion of a fixed vector. For a moving vector, we can show: ( d A dt ) F = ( d A dt ) R + Ω A GEF 2220: Dynamics p.36/315
Rotation If A = r, the position vector, then: So the acceleration is: ( d r dt ) F u F = u R + Ω r ( d u F dt ) F = ( d u F dt ) R + Ω u F = [ d dt (u R + Ω r)] R + Ω u F = ( d u R dt ) R + 2 Ω u R + Ω Ω r GEF 2220: Dynamics p.37/315
Rotation Rearranging: ( d u R dt ) R = ( d u F dt ) F 2 Ω u R Ω Ω r Two additional terms: Coriolis acceleration 2Ω u R Centrifugal acceleration Ω Ω r GEF 2220: Dynamics p.38/315
Centrifugal force Is the negative of centripetal acceleration Is the outward force we feel on a carousel If spinning too fast, we would fly off the earth If not, modifies the effective gravity GEF 2220: Dynamics p.39/315
Centrifugal force Ω x Ω x r g GEF 2220: Dynamics p.40/315
Centrifugal force Absorb into gravity: g = g Ω Ω r Example: What is the centrifugal force for a parcel at the Equator? GEF 2220: Dynamics p.41/315
Centrifugal force At the equator, we have: Ω Ω r = Ω (Ωr) = Ω 2 r with: r e = 6.378 10 6 m and: Ω = 2π 3600(24) sec GEF 2220: Dynamics p.42/315
Centrifugal force So: Ω 2 r e = 0.034 m/sec 2 This is much smaller than g = 9.8 m 2 /sec Only a minor change to absorb into g GEF 2220: Dynamics p.43/315
Cartesian coordinates Earth is spherical, so should use spherical coordinates We will primarily use Cartesian coordinates Simplifies the math The extra terms are usually unimportant at weather scales GEF 2220: Dynamics p.44/315
Cartesian coordinates Ω Ωcosθ Ω sinθ θ GEF 2220: Dynamics p.45/315
Coriolis force The rotation vector projects onto the local vertical and meridional directions: 2 Ω = 2Ωcosθ ĵ + 2Ωsinθ ˆk f y ĵ + f z ˆk So: 2 Ω u = (0,f y,f z ) (u,v,w) = (f y w f z v,f z u, f y u) GEF 2220: Dynamics p.46/315
Momentum equations Move Coriolis terms to the LHS: t u + u u + f yw f z v = 1 ρ t v + u v + f zu = 1 ρ t w + u w f yu = 1 ρ x p + ν 2 u y p + ν 2 v z p g + ν 2 w GEF 2220: Dynamics p.47/315
Coriolis force Example: What is the Coriolis force on a parcel moving eastward at 10 m/sec at 45 N? We have: f y = 2Ωcos(45) = (7.292 10 5 )(0.7071) = 5.142 10 5 sec 1 = f z 2Ω u = (0,f y,f z ) (u, 0, 0) = f z u ĵ + f y uˆk = (0, 5.142 10 4, 5.142 10 4 ) m/sec 2 GEF 2220: Dynamics p.48/315
Coriolis force So the force acts to the south and upwards Upward acceleration is negligible compared to gravity (g = 9.8 m/sec 2 ), so has little effect Horizontal acceleration is to the south Coriolis acceleration is most important in the horizontal Acts to the right in the Northern Hemisphere GEF 2220: Dynamics p.49/315
Coriolis force In the Southern hemisphere, θ < 0. Same problem, at 45 S: f y = 2Ωcos( 45) = 5.142 10 5 sec 1 = f z 2Ω u = (0, +5.142 10 4, 5.142 10 4 ) m/sec 2 Acceleration is to the north, to the left of the parcel velocity. GEF 2220: Dynamics p.50/315
Continuity δy x 0 y 0 z 0 δz δx GEF 2220: Dynamics p.51/315
Continuity Density flux through the left side: Through the right side: [ρu x (ρu) x ]δy δz 2 [ρu + x (ρu) x ]δy δz 2 GEF 2220: Dynamics p.52/315
Continuity So the net rate of change in mass is: (ρ x y z) = [ρu t x (ρu] x ) y z 2 [ρu + x (ρu) x 2 The volume δv is constant, so: ] y z = (ρu) x y z x t ρ = x (ρu) GEF 2220: Dynamics p.53/315
Continuity Taking the other sides of the box: ρ t = (ρ u) Can rewrite: (ρ u) = ρ u + u ρ. So: ρ t + u ρ + ρ( u) = 0 GEF 2220: Dynamics p.54/315
Continuity Can also derive using a Lagrangian box As the box moves, it conserves it mass. So: 1 M d dt ( M) = 1 ρδv d dt (ρδv ) = 1 ρ dρ dt + 1 δv dδv dt = 0 Expand the volume term: 1 δv dδv dt = 1 δx δx dt + 1 δy δy dt + 1 δz δz dt = u x + v y + w z GEF 2220: Dynamics p.55/315
Continuity So: 1 ρ dρ dt + u = 0 Same as before The change in density is proportional to the velocity divergence. If the volume changes, the density changes to keep the mass constant. GEF 2220: Dynamics p.56/315
Ideal Gas Law Five of the equations are prognostic: they describe the time evolution of fields. But one diagnostic relation. Relates the density, pressure and temperature GEF 2220: Dynamics p.57/315
Ideal Gas Law For dry air: p = ρrt where R = 287 Jkg 1 K 1 GEF 2220: Dynamics p.58/315
Moist air Law moist air, can write (Chp. 3): p = ρrt v where the virtual temperature is: T v T 1 e/p(1 ǫ) ǫ R d R v = 0.622 Hereafter, we neglect moisture effects GEF 2220: Dynamics p.59/315
Primitive equations Continuity: Ideal gas: ρ + u ρ + ρ u = 0 t Thermodynamic energy: p = ρrt dt c v dt + pdα dt = c dt p dt αdp dt = dq dt GEF 2220: Dynamics p.60/315
Thermodynamic equation q F GEF 2220: Dynamics p.61/315
First law of thermodynamics Change in internal energy = heat added - work done: dq dw = de Work is done by expanding against external forces: dw = Fdx = padx = pdv If dv > 0, the volume is doing the work GEF 2220: Dynamics p.62/315
First law of thermodynamics If volume has a unit mass, then: so: ρv = 1 dv = d( 1 ρ ) = dα where α is the specific volume. So: dq = p dα + de GEF 2220: Dynamics p.63/315
First law of thermodynamics Add heat to the volume, the temperature rises. The specific heat determines how much. At constant volume: c v dq dt v This is also the change in internal energy: c v = de dt v GEF 2220: Dynamics p.64/315
First Law of thermodynamics Joule s Law: e only depends on temperature for an ideal gas. So even if V changes: Result is the First Law: c v = de dt dq = c v dt + p dα GEF 2220: Dynamics p.65/315
First law of thermodynamics At constant pressure: c p dq dt p Volume expands keeping p constant. Requires more heat to raise the temperature. Rewrite the first law: dq = c v dt + d(pα) αdp GEF 2220: Dynamics p.66/315
First law of thermodynamics The ideal gas law is: p = ρrt = α 1 RT So: d(pα) = RdT Thus: dq = (c v + R)dT αdp GEF 2220: Dynamics p.67/315
First law of thermodynamics At constant pressure, dp = 0, so: dq dt p = c p = c v + R So the specific heat at constant pressure is greater than at constant volume. For dry air: c v = 717Jkg 1 K 1, c p = 1004Jkg 1 K 1 so: R = 287 Jkg 1 K 1 GEF 2220: Dynamics p.68/315
First law of thermodynamics So First Law can also be written: dq = c p dt αdp Obtain the thermodynamic equation by dividing by dt: dq dt = c dt v dt + pdα dt = c dt p dt αdp dt GEF 2220: Dynamics p.69/315
Basic balances Not all terms in the momentum equations are equally important for weather systems. Will simplify the equations by identifying primary balances (throw out as many terms as possible). Begin with horizontal momentum equations. GEF 2220: Dynamics p.70/315
Scaling General technique: scale equations using estimates of the various parameters. Take the x-momentum equation, without friction: t u + u x u + v y u + w z u + f yw f z v = 1 ρ x p U T U 2 L U 2 L UW D fw fu H P ρl GEF 2220: Dynamics p.71/315
Scaling Now use typical values. Length scales: L 10 6 m, D 10 4 m Horizontal scale is 1000 km, the synoptic scale. Large scale motion (not tornadoes). Velocities: U V 10m/sec, W 1cm/sec Winds are quasi-horizontal GEF 2220: Dynamics p.72/315
Scaling Pressure term: H P/ρ 10 3 m 2 /sec 2 Note typical horizontal difference. Time scale: T = L/U 10 5 sec Called an advective time scale ( 1 day). GEF 2220: Dynamics p.73/315
Scaling Coriolis terms: with (f y,f z ) = 2Ω(cosθ,sinθ) Ω = 2π(86400) 1 sec 1 Assume at mid-latitudes: f y f z = 2Ωsin(45) 10 4 sec 1 GEF 2220: Dynamics p.74/315
Scaling Plug in: t u + u x u + v y u + w z u + f yw f z v = 1 ρ x p U T U 2 L U 2 L UW D fw fu H P ρl 10 4 10 4 10 4 10 5 10 6 10 3 10 3 GEF 2220: Dynamics p.75/315
Geostrophy Keeping only the 10 3 terms: f z v = 1 ρ f z u = 1 ρ x p y p These are the geostrophic relations. Balance between the pressure gradient and Coriolis force. GEF 2220: Dynamics p.76/315
Geostrophy Fundamental momentum balance at synoptic scales Low pressure to left of the wind in Northern Hemisphere Low pressure to right in Southern Hemisphere But balance fails at equator, because f z = 2Ωsin(0) = 0 GEF 2220: Dynamics p.77/315
Geostrophy L p/ ρ H fu GEF 2220: Dynamics p.78/315
Geostrophy L H GEF 2220: Dynamics p.79/315
Geostrophy L fu GEF 2220: Dynamics p.80/315
Geostrophy Example: What is the pressure gradient required at the earth s surface at 45 N to maintain a geostrophic wind of 30 m/sec? f z = 2Ωsin(45) = 1.414 (7.27 10 5 ) sec 1 = 1.03 10 4 sec 1 p l = ρ 0f z u = (1.2 kg/m 3 )(1.03 10 4 sec 1 )(30 m/sec) = 3.7 10 3 N/m 3 =.37 kpa/100km GEF 2220: Dynamics p.81/315
Geostrophy Is a diagnostic relation Given the pressure, can calculate the horizontal velocities But geostrophy cannot be used for prediction GEF 2220: Dynamics p.82/315
Approximate horizontal momentum Must also retain the 10 4 terms: t u + u x u + v y u f zv = 1 ρ t v + u x v + v y v + f zu = 1 ρ x p y p The equations are quasi-horizontal: neglect vertical motion GEF 2220: Dynamics p.83/315
Other momentum balances Geostrophy most important balance at synoptic scales. But other balances possible. Consider purely circular flow: GEF 2220: Dynamics p.84/315
Other momentum balances Use cylindrical coordinates. Acceleration in radial direction is: t u r+(u r r +u θ r θ )u r u2 θ r fu θ = d dt u r u2 θ r fu θ = 1 ρ r p u 2 θ is the cyclostrophic term related to centripetal acceleration. GEF 2220: Dynamics p.85/315
Other momentum balances If constant circulation, dt d u = 0. Then: u 2 θ r + fu θ = 1 ρ r p Scaling: U 2 R fu H P ρr GEF 2220: Dynamics p.86/315
Other momentum balances So ratio of the first and second terms is: U fr ǫ This is the Rossby number GEF 2220: Dynamics p.87/315
Other momentum balances If ǫ 1, recover the geostrophic balance: fu θ = 1 ρ r p GEF 2220: Dynamics p.88/315
Other momentum balances If ǫ 1, the first term dominates. Happens at smaller scales and with stronger winds. A typical tornado at mid-latitudes has: U 30m/s, f = 10 4 sec 1, R 300m so that: ǫ = 1000 GEF 2220: Dynamics p.89/315
Cyclostrophic wind balance Have: or: u 2 θ r = 1 ρ r p u θ = ±( r ρ r p)1/2 Rotation does not enter. Circulation can go either way. The balance for bathtub vortices. GEF 2220: Dynamics p.90/315
Inertial oscillations Third possibility: there is no radial pressure gradient: then: u 2 θ r + fu θ = 0 u θ = fr Rotation is clockwise or anticyclonic in the Northern Hemisphere. GEF 2220: Dynamics p.91/315
Inertial oscillations The time for a fluid parcel to complete a loop is: 2πr = 2π u θ f = 0.5 day sinθ, Called the inertial period. Inertial oscillations are seen in the surface layer of the ocean, but are rarer in the atmosphere. GEF 2220: Dynamics p.92/315
Gradient wind balance Fourth possibility: all terms are important (ǫ = O 1 ). u 2 θ r + fu θ = 1 ρ r p Solve using the quadratic formula: u θ = 1 2 fr ± 1 2 (f2 r 2 + 4r ρ r p)1/2 = 1 2 fr ± 1 2 (f2 r 2 + 4r f u g ) 1/2 GEF 2220: Dynamics p.93/315
Gradient wind balance If u g < 0 (anticyclone), require: u g < fr 4 If u g > 0 (cyclone), there is no limit Wind gradients are stronger in cyclones than in anticyclones GEF 2220: Dynamics p.94/315
Gradient wind balance Alternately: u 2 θ r + fu θ = 1 ρ r p = fu g Then: u g u θ = 1 + u θ fr 1 + ǫ If ǫ = 0.1, the gradient wind estimate differs by 10 %. GEF 2220: Dynamics p.95/315
Gradient wind balance At low latitudes, where ǫ can be 1-10, the gradient wind estimate is more accurate. Geostrophy is symmetric to sign changes: no difference between cyclones and anticyclones The gradient wind balance is not symmetric to sign change. Stronger cyclones. GEF 2220: Dynamics p.96/315
Gradient wind balance L H p fv fv v r 2 v r 2 p GEF 2220: Dynamics p.97/315
Gradient wind balance Geostrophic motion has Coriolis and pressure gradient forces opposed. If cyclostrophic term large enough, gradient wind vortices can have the have pressure gradient and Coriolis forces in the same direction. GEF 2220: Dynamics p.98/315
Gradient wind balance L fv p v r 2 GEF 2220: Dynamics p.99/315
Gradient wind balance An anomalous low: low pressure with clockwise flow Usually only found near the equator GEF 2220: Dynamics p.100/315
Hydrostatic balance Now scale the vertical momentum equation Must scale: 1 ρ z p Vertical variation of pressure much greater than the horizontal variation: V P/ρ 10 5 m 2 /sec 2 GEF 2220: Dynamics p.101/315
Hydrostatic balance t w + u x w + v y w + w z w f yu = 1 ρ z p g UW L UW L UW L W 2 D fu V P ρd g 10 7 10 7 10 7 10 10 10 3 10 10 GEF 2220: Dynamics p.102/315
Static atmosphere Dominant balance between the vertical pressure gradient and gravity However, same balance if there no motion at all! Set all velocities to zero in the equations of motion: 1 ρ x p = 1 ρ y p = t ρ = dt dt = 0 GEF 2220: Dynamics p.103/315
Static atmosphere Only equations left: z p = ρg The hydrostatic balance. Also: p = ρrt Equations describe a non-moving atmosphere GEF 2220: Dynamics p.104/315
Static atmosphere Integrate the hydrostatic relation: p(z) = z ρg dz. The pressure at any point is equal to the weight of air above it. Sea level pressure is: p(0) = 101.325 kp a (1013.25mb) The average weight per square meter of the entire atmospheric column(!) GEF 2220: Dynamics p.105/315
Static atmosphere Say the temperature was constant (isothermal): This implies: z p = pg RT ln(p) = gz RT GEF 2220: Dynamics p.106/315
Static atmosphere So that: p = p 0 e z/h where the scale height is: H RT g Pressure decays exponentially with height GEF 2220: Dynamics p.107/315
Scaling Static hydrostatic balance not interesting for weather. Separate the pressure and density into static and non-static (moving) components: p(x,y,z,t) = p 0 (z) + p (x,y,z,t) ρ(x,y,z,t) = ρ 0 (z) + ρ (x,y,z,t) Assume: p p 0, ρ ρ 0 GEF 2220: Dynamics p.108/315
Scaling Then: 1 ρ z p g = 1 ρ 0 + ρ z (p 0 + p ) g 1 ρ 0 (1 ρ ρ 0 ) z (p 0 + p ) g = 1 ρ 0 z p + ( ρ ρ 0 ) z p 0 = 1 ρ 0 z p ρ ρ 0 g Neglect (ρ p ) GEF 2220: Dynamics p.109/315
Scaling Use these terms in the vertical momentum equation But how to scale? Vertical variation of the perturbation pressure comparable to the horizontal perturbation: 1 ρ 0 z p HP ρ 0 D 10 1 m/sec 2 GEF 2220: Dynamics p.110/315
Scaling Also: ρ 0.001 ρ 0 So: ρ ρ 0 g 10 1 m/sec 2 GEF 2220: Dynamics p.111/315
Scaling t w + u x w + v y w + w z w f yu = 1 ρ 0 z p ρ g ρ 0 10 7 10 7 10 7 10 10 10 3 10 1 10 1 GEF 2220: Dynamics p.112/315
Hydrostatic perturbations Dominant balance still hydrostatic, but with perturbations: z p = ρ g Vertical accelerations unimportant at synoptic scales. But lost the vertical velocity! Must calculate w indirectly. GEF 2220: Dynamics p.113/315
Coriolis parameter So all terms with f y are unimportant From now on, neglect f y and write f z simply as f: f 2Ωsin(θ) f y only important near the equator GEF 2220: Dynamics p.114/315
Pressure coordinates Can use the hydrostatic balance to simplify equations Constant pressure surfaces (in two dimensions): z p0 p0+dp dz dx x GEF 2220: Dynamics p.115/315
Pressure coordinates dp = p x dx + p z dz = 0 Substitute hydrostatic relation: dp = p x dx ρg dz = 0 So: p x z = ρg dz dx p ρ Φ x p GEF 2220: Dynamics p.116/315
Geopotential where: Φ z 0 g dz Instead of pressure at a certain height, think: Height of a certain pressure field GEF 2220: Dynamics p.117/315
Geopotential 4.2 km 520 hpa 4.1 km 510 hpa 4 km 500 hpa GEF 2220: Dynamics p.118/315
Geostrophy Removes density from the momentum equation! du dt fv = 1 ρ p x = Φ x Now the geostrophic balance is: fv = x Φ fu = y Φ GEF 2220: Dynamics p.119/315
Geostrophy 500 hpa Φ 1 Φ 2 Φ 3 GEF 2220: Dynamics p.120/315
Vertical velocities Different vertical velocities: w = dz dt ω = dp dt p 3 p 2 p 1 GEF 2220: Dynamics p.121/315
Geopotential Lagrangian derivative is now: d dt = t + dx dt x + dy dt y + dp dt p = t + u x + v y + ω p GEF 2220: Dynamics p.122/315
Continuity Lagrangian box: δv = δx δy δz = δx δy δp ρg with a mass: ρδv = δx δy δp/g GEF 2220: Dynamics p.123/315
Continuity Conservation of mass: 1 δm d dt δm = g δxδyδp d dt (δxδyδp ) = 0 g Rearrange: 1 δx δ(dx dt ) + 1 δy δ(dy dt ) + 1 δp δ(dp dt ) = 0 GEF 2220: Dynamics p.124/315
Continuity Let δ 0: u x + v y + ω p = 0 The flow is incompressible in pressure coordinates Much simpler to work with GEF 2220: Dynamics p.125/315
Diagnosing vertical motion Integrating the continuity equation: ω = p p ( x u + y v)dp If the top of the atmosphere, p = 0, so: ω = p 0 ( x u + y v)dp GEF 2220: Dynamics p.126/315
Vertical motion How does ω relate to the actual vertical velocity? ω = dp dt = t p + u x p + v y p + w z p Using the hydrostatic relation: For geostrophic motion: ω = dp dt = t p + u x p + v y p ρgw u x p + v y p = 1 ρf y p( x p) + 1 ρf x p( y p) = 0 GEF 2220: Dynamics p.127/315
Vertical motion So ω t p ρgw Also: t p 10hPa/day ρgw (1.2kg/m 3 ) (9.8m/sec 2 )(0.01m/sec) 100hPa/day GEF 2220: Dynamics p.128/315
Vertical motion So: ω ρgw This is accurate within 10 % in the mid-troposphere In the lowest 1-2 km, topography alters the balances At the surface: w s = u x z s + v y z s GEF 2220: Dynamics p.129/315
Vertical motion z (x,y) s GEF 2220: Dynamics p.130/315
Thermal wind Geostrophy tells us what the velocities are if we know the geopotential on a pressure surface What about the velocities on other pressure surfaces? Need to know the velocity shear Shear is determined by the thermal wind relation GEF 2220: Dynamics p.131/315
Thermal wind Can use geostrophy to calculate the shear between two pressure surfaces: v g (p 1 ) v g (p 0 ) = 1 f x (Φ 1 Φ 0 ) g f x Z 10 and: u g (p 1 ) u g (p 0 ) = 1 f y (Φ 1 Φ 0 ) g f y Z 10 GEF 2220: Dynamics p.132/315
Thermal wind where: Z 10 = 1 g (Φ 1 Φ 0 ) is the layer thickness between p 0 and p 1. Shear proportional to gradients of layer thickness GEF 2220: Dynamics p.133/315
Thermal wind Alternate expression. From the definition of the geopotential: dφ = g dz = dp ρ With the ideal gas law: Φ p = 1 ρ = RT p GEF 2220: Dynamics p.134/315
Thermal wind Now take the derivative wrt pressure of the geostrophic relation: But: So: p p (fv g = Φ x ) Φ x = x Φ p = R p T x p v g p = R f T x GEF 2220: Dynamics p.135/315
Thermal wind Or: v g ln(p) = R f T x also: u g ln(p) = R f T y Shear proportional to temperature gradient on p-surface GEF 2220: Dynamics p.136/315
Thermal wind If we know the velocity at p 0, can calculate it at p 1 Integrate between two pressure levels: v g (p 1 ) v g (p 0 ) = R f p1 p 0 T x d ln(p) = R f x p1 p 0 T d ln(p) GEF 2220: Dynamics p.137/315
Mean temperature Define the mean temperature in the layer bounded by p 0 and p 1 : T p1 p 0 T d(lnp) p1 p 0 d(lnp) = p1 p 0 T d(lnp) ln( p 1 p 0 ) Then: v g (p 1 ) v g (p 0 ) = R f ln(p 1 p 0 ) T x GEF 2220: Dynamics p.138/315
Thermal wind From before: v g (p 1 ) v g (p 0 ) = g f x Z 10 so: Z 10 = R g T ln(p 1 p 0 ) Layer thickness proportional to its mean temperature GEF 2220: Dynamics p.139/315
Layer thickness 9 p 2 p 1 v 2 7 8 v 1 6 GEF 2220: Dynamics p.140/315
Barotropic atmosphere What if temperature constant on all pressure surfaces? Then T = 0 no vertical shear Belocities have the same direction and magnitude with height All layers have equal thickness Stacked like pancakes GEF 2220: Dynamics p.141/315
Equivalent barotropic If temperature and geopotential contours are parallel, thermal wind is parallel to the wind itself Wind changes magnitude, but not direction with height Geostrophic wind increases with height if Warm high pressure Cold low pressure GEF 2220: Dynamics p.142/315
Equivalent barotropic Earth: zonal-average temperature decreases with latitude Poles are colder than equator the winds increase with height GEF 2220: Dynamics p.143/315
Jet stream 10 15 20 p GEF 2220: Dynamics p.144/315
Thermal wind Example: At 30N, the zonally-averaged temperature gradient is 0.75 Kdeg 1, and the average wind is zero at the earth s surface. What is the mean zonal wind at the level of the jet stream (250 hpa)? u g (p 1 ) u g (p 0 ) = u g (p 1 ) = R f ln(p 1 p 0 ) T y u g (250) = 287 2Ωsin(30) 250 ln( 1000 ) ( 0.75 1.11 10 5 ) = 36.8 m/sec m GEF 2220: Dynamics p.145/315
Baroclinic atmosphere Temperature and geopotential contours usually not parallel Geostrophic wind has a component normal to the temperature contours (isotherms) Produces geostrophic temperature advection Winds blow from warm to cold or vice versa GEF 2220: Dynamics p.146/315
Temperature advection v2 vt Z1+ δ Z Z1 v1 Warm ZT+ δz Cold ZT GEF 2220: Dynamics p.147/315
Temperature advection Warm advection veering Anticyclonic (clockwise) rotation with height Cold advection backing Cyclonic (counter-clockwise) rotation with height GEF 2220: Dynamics p.148/315
Summary Geostrophic wind parallel to geopotential contours Wind with high pressure to the right (North Hemisphere) Thermal wind parallel to thickness (mean temperature) contours Wind with high thickness to the right GEF 2220: Dynamics p.149/315
Divergence Continuity equation: dρ dt + ρ u = 0 or: 1 ρ dρ dt = u = ( u x + v y + w z ) Density changes due to flow divergence GEF 2220: Dynamics p.150/315
Divergence u < 0 u > 0 δu δx > 0 GEF 2220: Dynamics p.151/315
Divergence u > 0 u < 0 GEF 2220: Dynamics p.152/315
Divergence GEF 2220: Dynamics p.153/315
Example The divergence in a region is constant and positive: u = D > 0 What happens to the density of an air parcel? GEF 2220: Dynamics p.154/315
Example 1 ρ dρ dt = u = D dρ dt = ρd ρ(t) = ρ(0) e Dt GEF 2220: Dynamics p.155/315
Vorticity Central quantity in dynamics ζ u ζ = ( w y v z, u z w x, v x u y ) Most important at synoptic scales is vertical component: ζ = ζˆk = v x u y GEF 2220: Dynamics p.156/315
Vorticity y ζ = δu/ δy > 0 ζ = δu/ δy < 0 GEF 2220: Dynamics p.157/315
Vorticity ζ > 0 GEF 2220: Dynamics p.158/315
Example What is the vorticity of a typical tornado? Assume solid body rotation, with a velocity of 100 m/sec, 20 m from the center. In cylindrical coordinates, the vorticity is: ζ = 1 r rv θ r 1 r v r θ For solid body rotation, v r = 0 and v θ = ωr GEF 2220: Dynamics p.159/315
Vorticity Here ω =const. So: ζ = 1 r rv θ r = 1 r ωr 2 r = 2ω We have v θ = 100 m/sec at r = 20 m: So: ω = v θ r = 100 20 = 5rad/sec ζ = 10rad/sec GEF 2220: Dynamics p.160/315
2-D Shears Divergence: u x + v y Vorticity: v x u y GEF 2220: Dynamics p.161/315
2-D Shears Shear: u x v y Deformation: v x + u y GEF 2220: Dynamics p.162/315
Absolute vorticity Now add rotation. The velocity in the fixed frame is: u F = u R + Ω r So: ζ a = ( u + Ω r) = ζ + 2 Ω (as in solid body rotation example) GEF 2220: Dynamics p.163/315
Absolute vorticity Two components: u the relative vorticity 2Ω the planetary vorticity Vertical component dominates: ζ a ˆk = ( x v u) + 2Ωsin(θ) ζ + f y (ζ now refers to vertical relative vorticity) GEF 2220: Dynamics p.164/315
Absolute vorticity Scaling: ζ U L So: ζ f U fl = ǫ The Rossby number GEF 2220: Dynamics p.165/315
Absolute vorticity ǫ 1 Geostrophic velocities Planetary vorticity dominates Mid-latitude weather system GEF 2220: Dynamics p.166/315
Absolute vorticity ǫ 1 Cyclostrophic velocities Relative vorticity dominates Tornado GEF 2220: Dynamics p.167/315
Relative circulation Integrate the relative vorticity over an area of fluid: Γ ζ ˆn da = ( u) ˆn da = u dl ζ GEF 2220: Dynamics p.168/315
Circulation Derive equation for the circulation by integrating the momentum equations around a closed curve. First write momentum equations in vector form: d dt u + 2 Ω u = 1 ρ p + g + F Integrate around a closed area: d dt Γ + (2 Ω u) dl = p ρ dl + g dl + F dl GEF 2220: Dynamics p.169/315
Circulation Gravity vanishes because can write as a potential: g = gˆk = z ( gz) Φ g and the closed integral of a potential vanishes: Φ g dl = dφ g = 0 GEF 2220: Dynamics p.170/315
Absolute circulation So: d p dt Γ a = ρ dl + F dl where Γ a Γ + (2 Ω u) dl GEF 2220: Dynamics p.171/315
Absolute circulation It is possible to show that: (2 Ω u) dl = 2 Ω ˆn da So: d dt (2 Ω u) dl 2Ω da dt GEF 2220: Dynamics p.172/315
Kelvin s theorem If the atmosphere is barotropic (temperature constant on pressure surfaces): p ρ dl = 1 ρ p dl = 0 If atmosphere is also frictionless ( F = 0), then: d p dt Γ a = ρ dl + F dl = 0 The absolute circulation is conserved GEF 2220: Dynamics p.173/315
Example Move a circular region of air, with fixed area, north. Assume the air is initially at rest. What happens? GEF 2220: Dynamics p.174/315
Example Γ a (t 0 ) = Γ a (t f ) 0 + f 0 da = ζ da + f f da So: ζ da = (f f f 0 )da > 0 The relative vorticity increases GEF 2220: Dynamics p.175/315
Vorticity equation Horizontal momentum equations (p-coords): ( t + u x + v y + ω )u fv = p x Φ + F x ( t + u x + v y + ω p )v + fu = y Φ + F y Take x of the second, subtract y of the first GEF 2220: Dynamics p.176/315
Vorticity equation Find: ( t + u x + v y + ω p )ζ a = ζ a ( u x + v y ) + ( u p ω y v p ω x ) + ( x F y y F x) ζ a = ζ + f GEF 2220: Dynamics p.177/315
Vorticity equation So absolute vorticity can change due to three terms 1) The divergence term: ζ a ( u x + v y ) Divergence changes the vorticity, just like density GEF 2220: Dynamics p.178/315
Divergence Consider small area of air: δa = δx δy Time change in the area is: δa δt = δyδx δt + δxδy δt = δy δu + δx δv Relative change is the divergence: 1 δa δa δt = δu δx + δv δy GEF 2220: Dynamics p.179/315
Divergence So rewrite the divergence term: ( x u + y v)ζ a = ζ a A da dt So: d dt ζ a = ζ a A da dt d dt ζ aa = 0 This is Kelvin s theorem again! GEF 2220: Dynamics p.180/315
Vorticity equation 2) The tilting term: ( u p ω y v p ω x ) Difference in vertical motion can change the horizontal shear GEF 2220: Dynamics p.181/315
Tilting y w x GEF 2220: Dynamics p.182/315
Vorticity equation 3) The Forcing term: ( x F y y F x) Say frictional forcing: F x = ν 2 u, F y = ν 2 v GEF 2220: Dynamics p.183/315
Friction Then: ( x F y y F x) = ν 2 ( v x u y ) = ν 2 ζ Then: d dt (ζ + f) = ν 2 ζ GEF 2220: Dynamics p.184/315
Friction If f const.: Friction diffuses vorticity d dt ζ = ν 2 ζ Causes cyclones to spread out and weaken Can occur due to friction in the boundary layer GEF 2220: Dynamics p.185/315
Scaling ( t +u x +v y +ω p )ζ a = ζ a ( u x + v y )+( u p ω y v p ω x ) For synoptic scale motion: U 10m/sec ω 10hPa/day L 10 6 m p 100hPa f 0 10 4 sec 1 L/U 10 5 sec f y 10 11 m 1 sec 1 GEF 2220: Dynamics p.186/315
Scaling ζ U L 10 5 sec 1 So the Rossby number is: ǫ = ζ f 0 0.1 So: (ζ + f) f GEF 2220: Dynamics p.187/315
Scaling t ζ + u x ζ + v y ζ U2 L 2 10 10 ω p ζ Uω LP 10 11 v y f U f y 10 10 ( u p ω y v p ω x ) Uω LP 10 11 (ζ + f) ( u x + v y ) f ( u x + v y ) fu L 10 9 GEF 2220: Dynamics p.188/315
Scaling Divergence term unbalanced! To understand why, we need to look at the vertical velocities... GEF 2220: Dynamics p.189/315
Vertical velocity Continuity, with constant density: Scaling implies: or x u + y v + z w = 0 U L W D W U D L GEF 2220: Dynamics p.190/315
Vertical velocity Typically: But: D L 5km 1000km = 0.005 W U 1cm/sec 10m/sec = 0.001 The vertical velocity is 5 times smaller GEF 2220: Dynamics p.191/315
Vertical velocity Vorticity scales as: Divergence scales as: ζ = x v y u U L We have: D = x u + y v U L ζ U L 10m/sec 10 6 m = 10 5 sec 1 GEF 2220: Dynamics p.192/315
Vertical velocity From measurements: But: ζ 10 5 sec 1 D 10 6 sec 1! Divergence is 10 times smaller. Why? GEF 2220: Dynamics p.193/315
Ageostrophic velocities Write: u = u g + u a, v = v g + v a From the derivation of the gradient wind: This implies: u g u 1 + ǫ u a u g ǫ 0.1 GEF 2220: Dynamics p.194/315
Ageostrophic velocities So we can write: u = u g + ǫũ a, where ũ a = u a /ǫ. So the vorticity is: While the divergence is: ζ = x v g y u g + ǫ( v = v g + ǫṽ a xṽa yũa) D = x u g y v g + ǫ( xũa + yṽa) 1 f x ( Φ y ) + 1 f y ( Φ x ) +ǫ( xũa + yṽa) = ǫ( xũa + yṽa) GEF 2220: Dynamics p.195/315
Vertical velocities Also: w = D ǫ( + z xũa yṽa) So the divergence and the vertical velocity are order Rossby number Rotation suppresses vertical motion GEF 2220: Dynamics p.196/315
Scaling Thus the divergence estimate is ten times smaller than we had it before. So: (ζ + f) ( u x + v y ) f ( u x + v y ) ǫfu L 10 10 So now we have proper balance GEF 2220: Dynamics p.197/315
Scaling Retaining the 10 10 terms in the vorticity equation: ( t + u x + v ) (ζ + f) = f ( u y x + v y ) Is fairly accurate at synoptic scales Can even be used for prediction GEF 2220: Dynamics p.198/315
Forecasting Used for forecasts in the 1930 s and 1940 s Approach: Assume geostrophic velocities: u u g = 1 f v v g = 1 f Φ y Φ x GEF 2220: Dynamics p.199/315
Forecasting ζ ζ g = 1 f v g x u g y = 1 f ( 2 Φ x 2 + 2 Φ y 2 ) = 1 f 2 Φ The divergence vanishes: ( t + u g x + v g y ) (ζ g + f) = f ( u g x + v g y ) = 0 Implies ζ a is conserved following the horizontal winds Remember: on a pressure surface GEF 2220: Dynamics p.200/315
Forecasting Now only one unknown: Φ ( t + u g x + v g y ) (ζ g + f) = 0 becomes: ( t 1 f Φ y x + 1 f Φ x y ) (1 f 2 Φ + f) = 0 GEF 2220: Dynamics p.201/315
Forecasting Can write equation: or: t ζ g + u g ζ g + v g y f = 0 t ζ g = u g ζ g v g y f Can predict how ζ changes in time Then convert ζ Φ GEF 2220: Dynamics p.202/315
Example Assume the atmosphere is barotropic: no vertical shear Then Φ(x,y,p,t) = Φ(x,y,t) Solve: t ζ g = u g ζ g v g y f Need to specify y f GEF 2220: Dynamics p.203/315
Beta-plane approximation Assume the channel is centered at θ 0. Write f using a Taylor expansion: f(θ) = f(θ 0 ) + df dθ (θ 0) θ + 1 2 d 2 f dθ 2 (θ 0) θ 2 +... If range of latitudes is small, neglect the higher order terms: f f(θ 0 ) + df dθ (θ 0) θ GEF 2220: Dynamics p.204/315
Beta-plane approximation Change to Cartesian coordinates: Rθ y Then: f = f 0 + βy where: f 0 = 2Ωsin(θ 0 ), β = 2Ω R cos(θ 0) GEF 2220: Dynamics p.205/315
Beta-plane approximation So: v df dy = v y (f 0 + βy) = βv In addition, approximate: u g = 1 f v g = 1 f y Φ 1 f 0 y Φ x Φ 1 f 0 x Φ GEF 2220: Dynamics p.206/315
Example Assume the geopotential is: Φ = f 0 Uy + f 0 A sin(kx ωt)sin(ly) with: u g = 1 Φ = U la sin(kx ωt)cos(ly) f 0 y v g = 1 Φ = ka cos(kx ωt)sin(ly) f 0 x GEF 2220: Dynamics p.207/315
Example 1 Φ = sin(2x) sin(πy) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 GEF 2220: Dynamics p.208/315
Example 2 A sin(3x 2t) 1.5 1 0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 GEF 2220: Dynamics p.209/315
Example 2 A sin(3x+2t) 1.5 1 0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 GEF 2220: Dynamics p.210/315
Example The relative vorticity is: ζ g = 1 f 0 2 Φ = (k 2 + l 2 )A sin(kx ωt)sin(ly) Also need the derivatives: x ζ g = k(k 2 + l 2 )A cos(kx ωt)sin(ly) y ζ g = l(k 2 + l 2 )A sin(kx ωt)cos(ly) GEF 2220: Dynamics p.211/315
Example Collect terms: u x ζ v ζ = [U la sin(kx ωt)cos(ly)] y [k(k 2 + l 2 )A cos(kx ωt)sin(ly)] + [ka cos(kx ωt)sin(ly)] [l(k 2 + l 2 )A sin(kx ωt)cos(ly)] = Uk(k 2 + l 2 )A cos(kx ωt)sin(ly) GEF 2220: Dynamics p.212/315
Example Also: βv = βka cos(kx ωt)sin(ly) So: Also, since: t ζ = (U(k2 + l 2 ) β)ka cos(kx ωt)sin(ly) ζ g = 1 f 0 2 Φ = (k 2 + l 2 )A sin(kx ωt)sin(ly) GEF 2220: Dynamics p.213/315
Example Then: Equate both sides: t ζ = ω(k2 + l 2 )A cos(kx ωt)sin(ly) ω(k 2 + l 2 )A cos(kx ωt)sin(y) or: = (U(k 2 + l 2 ) β)ka cos(kx ωt)sin(ly) ω(k 2 + l 2 ) = (U(k 2 + l 2 ) β)k GEF 2220: Dynamics p.214/315
Example or: ω = Uk βk k 2 + l 2 The solution gives a specific value for the frequency! This is called a dispersion relation Relation between frequency, ω, and wavenumbers, (k, l) GEF 2220: Dynamics p.215/315
Phase speed If a travelling wave: ψ sin(kx ωt) the crests move with a phase speed: c x = ω k If ω > 0, waves move toward positive x (eastward) GEF 2220: Dynamics p.216/315
Phase speed c = 2/3 2 A sin(3x 2t) 1.5 1 0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 GEF 2220: Dynamics p.217/315
Phase speed We have: ω = Uk βk k 2 + l 2 so: c x = ω k = U β k 2 + l 2 GEF 2220: Dynamics p.218/315
Phase speed If U = 0, then: c x = β k 2 + l 2 All waves propagate westward! GEF 2220: Dynamics p.219/315
Westward propagation y=0 + GEF 2220: Dynamics p.220/315
Phase speed The wavelength in the x-direction is: λ x = 2π k and in the y-direction: λ y = 2π l GEF 2220: Dynamics p.221/315
Phase speed Say l = 0. Then: c x = β k 2 = βλ2 x 4π 2 The larger the wave, the faster it propagates The waves are dispersive If a collection of different waves initially, they separate by wavelength GEF 2220: Dynamics p.222/315
Phase speed If U 0, then: c x = ω k = U β k 2 + l 2 Longest waves go west while shorter waves are swept eastward by the zonal flow, U. If: k 2 + l 2 = β U the wave is stationary in the background flow GEF 2220: Dynamics p.223/315
Rossby waves These solutions are called Rossby waves Discovered by Carl Gustav Rossby (1936) Are observed in the atmosphere Stationary Rossby waves are important for long term weather patterns Will study more later GEF 2220: Dynamics p.224/315
Divergence Previously ignored divergence effects. But very important for the growth of unstable disturbances (storms) ( t + u x + v y ) (ζ + f) = d (ζ + f) = (ζ + f) ( u dt x + v y ) As air flows out of a divergent region, is deflected by the Coriolis force GEF 2220: Dynamics p.225/315
Divergence GEF 2220: Dynamics p.226/315
Divergence Consider flow with constant divergence: x u + y v = D > 0 d dt ζ a = ζ a ( u x + v y ) = Dζ a ζ a (t) = ζ a (0) e Dt GEF 2220: Dynamics p.227/315
Divergence So: ζ a = ζ + f 0 ζ f Divergent flow favors anticyclonic vorticity Vorticity approaches f, regardless of initial value GEF 2220: Dynamics p.228/315
Convergence GEF 2220: Dynamics p.229/315
Divergence Now say D = C d dt ζ a = ζ a ( u x + v y ) = Cζ a ζ a (t) = ζ a (0) e Ct ζ a ± But which sign? GEF 2220: Dynamics p.230/315
Divergence If the Rossby number is small, then: So: ζ a (0) = ζ(0) + f f > 0 ζ + Convergent flow favors cyclonic vorticity Vorticity increases without bound Why intense storms are almost always cyclonic GEF 2220: Dynamics p.231/315
Summary The vorticity equation is approximately: or: d (ζ + f) = (ζ + f) ( u dt x + v y ) d dt ζ + v df dy = (ζ + f) ( u x + v y ) Vorticity changes due to meridional motion Vorticity changes due to divergence GEF 2220: Dynamics p.232/315
Forecasting t ζ = u ζ v f (ζ + f)( u y x + v y ) Write in difference form: ζ(x,y,t 1 ) ζ(x,y,t 0 ) t = u(x,y,t 0 ) ζ(x,y,t 0 ) v(x,y,t 0 ) y f(y) [ζ(x,y,t 0)+f(y)][ u(x,y,t 0) x + v(x,y,t 0) ] y GEF 2220: Dynamics p.233/315
Forecasting or: ζ g (x,y,t 1 ) = ζ g (x,y,t 0 ) ( t)u g (x,y,t 0 ) ζ g (x,y,t 0 ) ( t)v g (x,y,t 0 ) y f(y) [ t) (ζ(x,y,t 0) + f(y)] ( u(x,y,t 0) x + v(x,y,t 0) ) y With fields at t 0, can predict vorticity at t 1 GEF 2220: Dynamics p.234/315
Forecasting Method: Obtain Φ(x,y,t 0 ) from measurements on p-surface Calculate u g (t 0 ), v g (t 0 ), ζ g (t 0 ) Calculate ζ g (t 1 ) Invert ζ g to get Φ(t 1 ) Start over Obtain Φ(t 2 ), Φ(t 3 ),... GEF 2220: Dynamics p.235/315
Inversion ζ g = 1 f ( 2 Φ x 2 + 2 Φ y 2 ) 2 Φ = fζ g Poisson s equation Need boundary conditions to solve GEF 2220: Dynamics p.236/315
Inversion Simple example: channel, with no flow at boundaries. Let: ζ = sin(3x)sin(πy) x = [0, 2π], y = [0, 1] So: 2 2 x2φ + y2φ = sin(3x)sin(πy) GEF 2220: Dynamics p.237/315
Inversion Try a particular solution: Φ = Asin(3x)sin(πy) This solution works in a channel, because: Φ(x = 2π) = Φ(x = 0) Also, at y = 0, 1: v = 1 f 0 Φ x = 0 GEF 2220: Dynamics p.238/315
Inversion Substitute into equation: 2 2 x2φ + y 2Φ = (9 + π2 )Asin(3x)sin(πy) = sin(3x)sin(πy) So: A = 1 9 + π 2 So we have Φ, and we can proceed GEF 2220: Dynamics p.239/315
Inversion Generally though, the vorticity is complicated Solve Poisson s equation numerically Often takes the most time during a calculation GEF 2220: Dynamics p.240/315
Barotropic potential vorticity Imagine atmosphere as a layer with constant density. Lies between two boundaries at z = z 1,z 2 (e.g. the surface and the tropopause) The continuity equation is: dρ dt + ρ( u) = 0 If density constant, then: ( u) = u x + v y + w z = 0 GEF 2220: Dynamics p.241/315
Barotropic potential vorticity So: u x + v y = w z Thus the vorticity equation can be written: ( t + u x + v w ) (ζ + f) = (ζ + f) y z With constant density: z u = z v = 0 GEF 2220: Dynamics p.242/315
Barotropic potential vorticity So we can integrate vertically: z2 z1 ( t + u x + v ) (ζ + f)dz = y h( t + u x + v y ) (ζ + f) = (ζ + f) (w(z 2) w(z 1 )) where h = z 2 z 1. Note that w = Dz/Dt. Thus: w(z 2 ) w(z 1 ) = d dt (z 2 z 1 ) = dh dt GEF 2220: Dynamics p.243/315
Barotropic potential vorticity So: or: h d dh (ζ + f) = (ζ + f) dt dt 1 ζ + f d dt (ζ + f) 1 h dh dt = 0 d dt ln(ζ + f) d dt lnh = 0 d dt lnζ + f h = 0 GEF 2220: Dynamics p.244/315
Barotropic potential vorticity Thus: Therefore: d dt (ζ + f h ) = 0 ζ + f h = const. the barotropic potential vorticity (PV) The PV takes into account changes in layer thickness If h increases, either ζ or f must also increase GEF 2220: Dynamics p.245/315
Layer potential vorticity GEF 2220: Dynamics p.246/315
Alternate derivation Consider a column between z 1 and z 2 As it moves, conserves its mass: So: or: d dt (ha) = 0 A dh dt + hda dt = 0 1 h dh dt + 1 A da dt = 0 GEF 2220: Dynamics p.247/315
Alternate derivation The relative change in area is the divergence: 1 A da dt = 1 δx δy d dt (δxδy) = 1 (δuδy + δxδv) δx δy So: = δu δx + δv δy u x + v y 1 h dh dt = u x + v y GEF 2220: Dynamics p.248/315
Alternate derivation Substitute for the divergence in the vorticity equation: d (ζ + f) = (ζ + f)( u dt x ) + v y which leads to: = ζ + f h d dt ζ + f h dh dt = 0 GEF 2220: Dynamics p.249/315
Adiabat Same idea applies to motion on an adiabatic The thermodynamic energy equation is: c p dt αdp = dq From the ideal gas law: So: pα = RT c p dt RT p dp = dq GEF 2220: Dynamics p.250/315
Adiabat dt c p T Rdp p = c p dlnt R dlnp = dq T If no heating, dq = 0, and: c p dlnt = R dlnp Defines an adiabat: a surface on which a parcel can freely move without external heating GEF 2220: Dynamics p.251/315
Potential temperature Heating changes the temperature and the pressure Define a temperature to remove the pressure effect Imagine a parcel with pressure p and temperature T Compress or expand adiabatically to a standard pressure: p 0 = 100 kpa = 1000mb This is potential temperature: c p lnt R lnp = c p lnθ R lnp 0 GEF 2220: Dynamics p.252/315
Potential temperature Rearranging and taking the exponential of both sides: θ = T ( p 0 p )R/c p Potential temperature depends on T and p If zero heating, a parcel conserves its potential temperature Call a surface with constant potential temperature an isentropic surface GEF 2220: Dynamics p.253/315
Layer potential vorticity Flow between two isentropic surfaces trapped if zero heating So mass in a column between two surfaces is conserved: From the hydrostatic relation: Aδz = const. Aδp ρg = const. where δp is the spacing between surfaces GEF 2220: Dynamics p.254/315
Layer potential vorticity θ + δθ δρ θ GEF 2220: Dynamics p.255/315
Layer potential vorticity Relate δp to δθ using the static stability: δp = ( θ p ) 1 δθ The stronger the stratification, the smaller the pressure difference between surfaces. So: A( θ δθ ) 1 p g = const. GEF 2220: Dynamics p.256/315
Layer potential vorticity We have ρ const. between the surfaces. Also δθ and g are constant. So: or: d dt [A( θ p ) 1 ] = 0 ( θ p ) 1 d dt A + A d dt ( θ p ) 1 = ( θ p ) 1 d dt A A( θ p ) 2 d dt ( θ p ) GEF 2220: Dynamics p.257/315
Layer potential vorticity Rearranging: From before: so: 1 A d dt A ( θ p ) 1 d dt ( θ p ) = 0 1 A d dt A = x u + y v ( θ p ) 1 d dt ( θ p ) = u x + v y GEF 2220: Dynamics p.258/315
Layer potential vorticity Substituting again from the vorticity equation: or: 1 ζ + f d (ζ + f) + ( θ d dt p ) 1 dt ( θ p ) = 0 d [(ζ + f) θ dt p ] = 0 This is the isentropic potential vorticity, first derived by Ertel (1942) GEF 2220: Dynamics p.259/315
Layer potential vorticity Remember: ζ evaluted on potential temperature surface Very useful quantity: can label air by its PV Can distinguish air in the troposphere which comes from stratosphere GEF 2220: Dynamics p.260/315
Layer potential vorticity Can be used for prediction: [(ζ + f) θ t p ] = (u x + v )[(ζ + f) θ y p ] Can predict how the PV changes on the isentrope Then invert the PV to determine the winds Not as accurate as solving the full primitive equations, but very useful for understanding/visualization GEF 2220: Dynamics p.261/315
Planetary boundary layer z=0 GEF 2220: Dynamics p.262/315
Turbulence There is a continuum of eddy scales Largest resolved by our models, but the smallest are not. GEF 2220: Dynamics p.263/315
Boussinesq equations Assume we can split the velocity into a mean (over some period) and a perturbation: u = u + u Use the momentum equations with no friction: t u + u u x + v u y + w u z fv = 1 ρ t v + u v x + v v y + w v z + fu = 1 ρ x p y p GEF 2220: Dynamics p.264/315
Boussinesq equations Assume density in the boundary layer approximately constant, so that: x u + y v + z w = 0 Substitute the partitioned velocities into the momentum equations and then average: t (u + u ) + (u + u ) x (u + u ) + (v + v ) y (u + u ) f(v + v ) +(w + w ) z (u + u ) = 1 ρ x (p + p ) GEF 2220: Dynamics p.265/315
Boussinesq equations Now average. Note that: so: u + u = u t u + u x u + u x u + v y u + v y u + +w z u + w z u + fv = 1 ρ x p GEF 2220: Dynamics p.266/315
Boussinesq equations Because of the continuity equation, we can write: So: u x u + v y u + w z u = x u u + y u v + z u w t u + u x u + v y u + w z u fv = = 1 ρ x p ( x ρu u + y u v + z u w ) GEF 2220: Dynamics p.267/315
Boussinesq equations Similarly: t v + u x v + v y v + w z v + fu = = 1 ρ y p ( x ρv u + y v v + z v w ) Terms on the RHS are the eddy stresses GEF 2220: Dynamics p.268/315
PBL equations Assume the eddy stresses don t vary horizontally. Then: t u + u x u + v y u + w z u fv = 1 ρ x p z u w t v + u x v + v y v + w z v + fu = 1 ρ y p z v w GEF 2220: Dynamics p.269/315
PBL equations Outside the boundary layer, assume geostrophy. In the layer, we have geostrophic terms plus vertical mixing. So turbulence breaks geostrophy: fu = 1 ρ fv = 1 ρ x p z u w = fv g z u w y p z v w = fu g z v w GEF 2220: Dynamics p.270/315
PBL equations But too many unknowns! : u, v,u,v,w Must parameterize the eddy stresses. Two cases: Stable boundary layer: stratified Unstable boundary layer: convective mixing GEF 2220: Dynamics p.271/315
Convective boundary layer Layer is vertically mixed by convection Temperature and velocity are constant with height From measurements, surface fluxes: u w s = C d V u, v w s = C d V v C d is the "drag coefficient" and V (u 2 + v 2 ) 1/2 GEF 2220: Dynamics p.272/315
Convective boundary layer Integrate vertically: h 0 f(v v g )dz = h 0 z u w dz Mixing vanishes outside of the layer, so: u w h = 0 Thus: fh(v v g ) = u w s GEF 2220: Dynamics p.273/315
Convective boundary layer With the parameterization: fh(v v g ) = C d V u Likewise: fh(u u g ) = C d V v GEF 2220: Dynamics p.274/315
Convective boundary layer Say v g = 0; then: v = C d fh V u, u = u g C d fh V v Solving equations not so simple because V = u 2 + v 2 But can use iterative methods GEF 2220: Dynamics p.275/315
Convective boundary layer If u > 0, then v > 0 L u b u g Flow down the pressure gradient GEF 2220: Dynamics p.276/315
Stable boundary layer Now assume no large scale vertical mixing Wind speed and direction can vary with height Specify turbulent velocities using mixing length theory: where l > 0 if up. u = l z u GEF 2220: Dynamics p.277/315
Mixing length u(z) l GEF 2220: Dynamics p.278/315
Stable boundary layer So: u w = w l z u Assume the vertical and horizontal eddy scales are comparable where again V = u 2 + v 2 Notice w > 0 if l > 0. w = l z V GEF 2220: Dynamics p.279/315
Stable boundary layer So: Same argument: ρu w = (ρl 2 z V) z u A z z u ρv w = A z z v where A z is the eddy exchange coefficient Depends on the size of turbulent eddies and mean shear GEF 2220: Dynamics p.280/315
Ekman layer An important example of a stably stratified PBL Studied by Swedish oceanographer V. W. Ekman (1905) Assume A z is constant Simplest version of a stratified layer GEF 2220: Dynamics p.281/315
Ekman layer So we have: A z 2 z 2u + f(v v g) = 0 A z 2 z 2v f(u u g) = 0 For boundary conditions, use the no-slip condition : u = 0,v = 0 at z = 0 GEF 2220: Dynamics p.282/315
Ekman layer Assume far from the surface, the velocities approach their geostrophic values: u u g,v v g z Assume the geostrophic flow is zonal and independent of height: u g = U, v g = 0 GEF 2220: Dynamics p.283/315
Ekman layer Boundary layer velocities vary only in the vertical: u = u(z), v = v(z), w = w(z) From continuity: x u + y v + z w = z w = 0. With a flat bottom, this implies: w = 0 GEF 2220: Dynamics p.284/315
Ekman layer The system is linear, so can decompose the horizontal velocities: u = U + û v = 0 + ˆv Then: 2 fˆv = A z z 2û 2 fû = A z z 2 ˆv. GEF 2220: Dynamics p.285/315
Ekman layer Boundary conditions: Introduce a new variable: û = U, ˆv = 0 at z = 0 Then: χ û + iˆv 2 z 2χ = i f A z χ. GEF 2220: Dynamics p.286/315
Ekman layer The solution is: if if χ = A exp( z) + B exp( z) A z A z Constants A and B are possibly complex. Rewrite: then: i = exp( iπ 2 ) = exp(iπ 4 ) = 1 2 + i 2 χ = A exp( z δ E )exp(i z δ E ) + B exp( z δ E )exp( i z δ E ), GEF 2220: Dynamics p.287/315