2. Baroclinic Instability and Midlatitude Dynamics

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1 2. Baroclinic Instability and Midlatitude Dynamics Midlatitude Jet Stream Climatology (Atlantic and Pacific) Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/1

2 Eddies What sets the scale of eddies? What is the mechanism for their development? Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/2

3 Instabilities Rotating fluid adjusts to geostrophic equilibrium: has potential energy available for conversion to other forms. Examine small disturbances: do dynamical constraints allow the disturbances to draw on APE? Basic midlaitude flow in lower atmosphere is jetstream with both horizontal and vertical mean-flow shears. wind temperature Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/3

Barotropic instability: wave instability associated with horizontal shear. Extracts KE from mean flow. Baroclinic instability: wave instability associated with vertical shear.

4 Barotropic instability: wave instability associated with horizontal shear. Extracts KE from mean flow. Baroclinic instability: wave instability associated with vertical shear. Converts PE associated with mean horizontal temp gradient that must exist to provide thermal wind balance for vertical shear. Wave instabilities important for synoptic-scale meteorology are zonally asymmetric perturbations (eddies) to zonally symmetric flow field. Baroclinic instabilities also observed in ocean - often more complex since the flow is rarely purely zonal. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/4

5 But, having PE available in flow is not sufficient for instability - rotation tends to inhibit release of PE. Analogy with shear instability of 2-D vortex dynamics always KE available, but flow isn t always unstable. How can the energy be released? Consider simple model that contains the important feature vertical shear (and hence horizontal density contrasts). Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/5

6 Laboratory model Warm Equator Cold Pole Warm Equator Formation of eddies Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/6

7 The Eady problem Eady problem: disturbances to westerly sheared jet, meridional temperature gradient. Stability properties? A simple paradigm for baroclinic instability. Analyse structure (eigenfunctions) and growth rates (eigenvalues) for unstable modes in simplest possible model that satisfies necessary conditions for instability. The model (Eady, 1949; see also Charney 1947): Boussinesq approximation (basic state density constant) f-plane approximation, β = u / z = Λ = constant rigid lid at z =, H 2 N taken to be constant Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/7

8 Constant rotation f f. Simplifications not realistic for atmosphere, or (except constant density) for ocean. But model just rich enough to expose fundamental nature of the instability. [Infinite region of large static stability 2 N in place of upper lid gives only qualitative changes in results.] Use quasi-geostrophic potential vorticity (QGPV) Dg equation: q = where Dt thermodynamic equation: q 2 = ψ + f N D 2 g ψ N w 2 ψ, and the z = Dt z f. For basic state with ψ = Λ zy, potential vorticity is zero q =. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/8

9 Linearised QGPV equation is: u + q =, t x implies QGPV of disturbance q is zero in interior. Thus dynamics set by boundary conditions. Without upper boundary flow would be stable. Linearised thermodynamic equation at a boundary where w = is: f f u u ψ ψ + t x 2 = z 2 N N x z. Vertical shear at upper & lower boundaries implies temperature gradient (in cross-flow direction). Advection of temperature field at upper & lower boundaries must therefore be taken into account. In Eady problem evolution of flow controlled by temperature at upper & lower boundaries Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/9

10 Eady edge waves Consider lower boundary only. Equations become: f ψ xx + ψ yy + ψ 2 zz f = and Λf ψ 2 z ψ 2 x =. N t N N ikx+ ily iωt Look for solutions with ψ = Re[ ϕ( z) e ]. kλf Dispersion reln: ω = N( k + l ) 2 2 1/ 2 Consider upper boundary only. kλf Dispersion reln: ω = + kλh 2 2 1/ 2 N( k + l ) eastward phase speed westward phase speed relative to flow Warm anomalies on lower (upper) level: (anti)cyclonic flow, propagate to east (west relative to mean flow). C C C But no heat is transported poleward and no energy is released. Lower boundary Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/1

11 ikx+ ily iωt Both boundaries included: ψ = Re[ ϕ( z) e ] 2 2 1/ 2 ϕ( z) = Acoshκ z + Bsinhκ z where κ = ( k + l ) ψ zt Λ ψ x = Boundary conditions give A, B. ψ zt + Λ Hψ x = N f 1 Dispersion relation: ω = kλ H ± 2 1 cothκ H 1 α = + 4 κ H κ H 2 2 α where ω real or complex depending on sign of α. When α < flow is baroclinically unstable. When α = α c = flow is neutrally stable. Hence instability requires α < α or ( ) αc k + l < f / N 5.76 / L where R L = NH / f 1km is Rossby radius of R deformation. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/11 c

12 1. Deep domain/ short waves ( κ H >> 1) 1/ 2 k H ω κ H k H Λ ± + Λ ± 2 2 κ H ω ω + kλ κ κλh kλ κ eastward wave (lower boundary) westward wave relative to flow (upper boundary) 2. Shallow domain /long waves ( κ H << 1) α <, ω is complex instability. Maximum growth rate for smallest κ, i.e. when l =. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/12

13 Insights from the Eady problem f 1. Wavenumber k for no growth / LR NH = 2. In channel with rigid walls, instability occurs if channel width NH > 1.31 = 1.31L R. Current must be at f least as broad as deformation radius for instability. 3. Maximum growth rate occurs at l = (no cross-flow structure). Max growth rate.31 f NΛ. f 4. Wavenumber k for max growth / LR NH =. 5. Length scale (quarter wavelength) of fastest growing mode just under L R. (Wavelength λ is λ = 2 π / k ). Thus expect to see unstable disturbances with scales of order the deformation radius. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/13

This is fundamental reason why both atmosphere and ocean are filled with perturbations with horizontal spatial scales of order the deformation radius.

14 This is fundamental reason why both atmosphere and ocean are filled with perturbations with horizontal spatial scales of order the deformation radius. Atmosphere: NH LR f = ~ 1km & f Λ ~ 5 1 s N 5 1 e-folding time for fastest growing waves: Wavelength of fastest growing waves: 18 hours 4 km Seems to be consistent with midlatitude cyclones. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/14

15 Phase relation For growing mode, westward phase tilt with height Velocity pattern increases temperature perturbation & resists tilting by shear Warmest air moves poleward at the steering level Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/15

16 Density/heat transport by the growing wave To extract energy from system, must be net poleward transfer of heat (i.e. v must be correlated with ρ ) For wave on boundary, correlation v ρ exactly zero. For growing Eady wave v ρ <, v T >. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/16

17 Laboratory experiment revisited Pole Equator Poleward transport of heat Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/17

18 Barotropic/baroclinic instabilities APE not sufficient for instability. What are conditions? Baroclinic/barotropic instabilities involve Rossby wave dynamics (PV or surface temp advection) f q y = β u yy u zρ ρ N Require counter-propagating Rossby waves. PV contours z QGPV gradient in basic state ¼ wavelength offset Necessary: q y, U z upper boundary, ½ wavelength offset U lower z boundary must take both signs somewhere in flow. Contrasting signs in vertical baroclinic instability in horizontal barotropic instability Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/18

19 Baroclinic instability releases PE from mean flow Barotropic instability releases KE from mean flow [but note wave need not be unstable to release energy from mean flow] What limits growth? Instability of shear layer: unstable growth ceases when layer rolls up into vortices pre-existing vorticity gradients have been considerably distrupted. Baroclinic instability: growth ceases when temperature gradient at lower boundary is stirred up [but not necessarily the temperature gradient in the interior] Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/19

20 Examples 1. interior q y >, du dz bottom > midlatitude circulation of atmosphere (with positive q y concentrated at tropopause) 2. interior q y >, du dz top < westward flowing part of wind-driven ocean circulation 3. interior q y > in some regions, < in others SH summer mesosphere? (Plumb 1983) [possible explanation for 2-day wave?] Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/2

21 Summary Atmosphere must transport heat poleward to maintain temperature gradient: key role in climate. Angular momentum must be transported poleward since westerly (easterly) winds at surface in midlatitudes (tropics). Midlatitude baroclinic eddies transport heat & angular momentum poleward. Copyright 26 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 2/21

3. Midlatitude Storm Tracks and the North Atlantic Oscillation

3. Midlatitude Storm Tracks and the North Atlantic Oscillation Copyright 2006 Emily Shuckburgh, University of Cambridge. Not to be quoted or reproduced without permission. EFS 3/1 Review of key results