Tues 20v08

The collection of papers from the Savannah Georgia meeting on Jets and Annular Modes in Atmospheres and Oceans is linked above.

EP fluxes and PV fluxes.

• Edmon, Hoskins & McIntyre, Eliassen-Palm cross-sections for the troposphere, JAS 1981, which is quite well documented in Vallis' text.
• Wernli & Davies, A Lagrangian based analysis of extratropical cyclones, I. QJ Roy.Met.Soc 1997.

  We looked at Edmon's et al. and Vallis' treatment of EP fluxes: interaction between the transient or stationary waves/eddies with the zonal mean atmosphere. The life-cycle of baroclinic instability is one interesting example (there are many more less famous examples). The horizontal and vertical parts of the EP flux represent, respectively, pressure drag on vertical and nearly horizontal material surfaces: form drag. Yet the horiz. part is more usually described as Eulerian momentum flux (merid. flux of zonal mom.). The vertical part involving the x-average of v'θ'/θ_p converts to
  ρf(v'ξ')_z = (p'_xξ')_z = -(p'ξ'_x)_z
  where ξ is the vertical coordinate of an isentropic surface; we remember this expression as pressure x slope of the isentropic surface in the x-direction, which is just that zonal pressure force. The z-derivative is there because there are isentropic surfaces above and below you, and the net force is the difference between the pressure force exerted on the wavy surfaces above and below. See Vallis Sec. 3.5. It is simpler using a multi-layer model of the stratification, in which case the expression is ρf(v'h'), where h' is the thickness of the isopycnal layer between its two interfaces.

  The divergence of the EP flux F drives the zonal mom. equation (Edmon et al. 2.3a), and the thermal wind balance (2.3c) is in force (being a horiz. vorticity equation, it is the result of geostrophic adjustment after external forces are diabatic heating push us away from geostrophic balance). But the fv* term in the zonal mom equation needs to be solved for too. So cross-differentiating the zonal mom eqn and the pot. temperature equation gives the new equation (following page) for the residual streamfunction ψ* whose cross derivatives in the meridional plane give you the meridional overturning circulation for v* and ω*. It is a stretched Laplace operator in y and p...meridional and vertical directions, forced by (&nabla*F)_p. So the MOC is a Poisson eqn (forced Laplace eqn) driven by EP flux and diabatic heating and mechanical forcing. Together these two equations give you both the zonal mom acceleration and the MOC.

  PV flux is equal to the divergence of the EP flux in the QG zonally averaged framework. Two differences: PV is finerin length scale than momentum, hence more difficult to map. Yet PV flux and conservation statements are fully 3-dimensional whereas EP flux is defined for the zonal average of the atmosphere. 3D EP fluxes have been developed, to study things like storm tracks and difflent/confluent jet stream regions...but they are rather complicated.

  Yet the greatest strength of PV analysis is its tracer-like property: in absence of forcing or dissipation PV is conserved following fluid parcels. The large-scale 'geography' of PV set by the spherical Earth, the solid Earth topography and the mean circulation is well observed. Stirring the fluid with waves or instabilities leads to predictable PV fluxes and interesting limiting cases like PV homogenization; this is far less true with EP fluxes because pressure forces can be so unpredictable.

  Both EP and PV are important to wave propagation. We are used to thinking that wave packets move with the group velocity and carry their wave energy with them. But in GFD these waves interact with the large-scale state of the fluid, inducing mean flow. In presence of a large-scale flow they do not conserve their wave energy, but can gain or lose energy from the mean flow. We looked at wave action,
  E/(ω-Uk) = E/(frequency seen by observer riding on the mean flow)
  which is the conserved property, instead of wave energy, for waves in a medium with mean flow that varies in space (and or time). It is simply equal to pseudo mom or 'wave activity', divided by the x-wavenumber k, in the basic zonally averaged EP world. It moves with the group velocity and expresses the mom flux by the wave packet, as the wave packet propagates. A. Einstein first discovered it for a pendulum whose length is changing in time...or a grad student swinging on a swing.

  While this subject is complex, it has underlying principles which are straightforward. We remarked in our barotropic Rossby wave generation problem that when you push on a fluid with a wavy wall, exerting a force Fo moving at velocity c, you create momentum (or more generally impulse) at rate Fo kg m/sec² and also create energy in the fluid at rate cFo kg m²sec^-3. Thus we come to see that waves have a momentum flux equal to their energy flux divided by the phase speed c. This helps to understand expressions for pseudomomentum and wave action.
  

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  Week 7. 12v-16v08
  
  

  Tues 13v08 Energy equations!
  The sections in Vallis to read on energy balance are:
  1.10 (uniform density and stratified fluids)
  3.10 (APE)
  5.6 (QG energetics) for comparison, also see
  2.4.3 (Boussinesq energetics)
  Gill's Atmosphere-Ocean Dynamics is still an excellent resource; his discussion of APE in section 7.8 is worth looking at.

  Looking at constant density and fully stratified/compressible equations, noting especially the role of the Bernoulli function in giving a simple energy flux (plus rate working by pressure forces, which transmit energy from one fluid parcel to the next), the role of the geopotential Φ (approximately Φ=gz) as a potential for the gravity + centrifugal body force, and hence as an absolute potential energy per unit mass. Work against gravity changes potential energy, through (ρ w gz).

  With stratification and compressibility we have tidy expressions for total energy = KE + PE +IE. Internal energy (IE) communicates with external mechanical energy (KE+PE) through the compression term, ρ&nabla*u. The 1st law of thermodynamics is the relevant internal energy equation (though it is not very clear in thermodynamics text discussions of this 1st law how to deal with external KE...it is largely ignored by the 1st law, but is lurking in the microscopic nature of interrnal thermal energy: at least for simple molecules it is mechanical energy in disguise). Again KE and PE communicate through the correlation of vertical velocity and density. APE (available PE) uses as a zero-reference state the fluid adiabatically adjusted so that density is constant along geopotential horizons. Recall that a 'resting rotating fluid' is one in which pressure, density, potential density are all constant along Φ=constant surfaces ('horizons'). Very often APE is expressed as a quadratic expression gρ'²/ρ_z where ρ' is density perturbation relative to this rest state.

  Energy flux and 'heat flux' become subtle with compressibility: the atmospheric meridional overturning circulation flows cold air northward and warmer air southward (due to the huge lapse rate) yet dry static energy flux or heat flux based on potential temperature is directed poleward.

  The atmosphere/ocean is a giant heat engine running at low efficiency (the temperatures of regions radiating to space and accepting radiation from space are not too different on the absolute Kelvin scale). Its energy flow is partitioned into 3 roughly equal streams: dry static energy (atmos) and its analog in the ocean, 'sensible heat flux', and a 3d, atmos/ocean coupled mode, the latent-heat transport in atmos, returned as excess fresh water in the oceans. Each stream has a peak ampltiude of about 2 petawatts (2x10¹⁵ watts).

  We looked at the conservation equation involving 1/2d/dt(ρ²) which involves two flux terms: ρ'w'dρ/dz and ρ'v'dρ/dy. Think of these as the vector heat flux dotted with the vector zonal mean density gradient. From our KE equation we know that to release APE ρ'w' must be <0. dρ/dz <0 too so for the eddy APE to grow we must have ρ'v' dρ/dz <0 which means eddy heat flux from warm latitudes toward cold latitudes. And this horizontal flux must be big enough to orient the vector density flux in the 'wedge of instability', more horizontal than the isentropic sloping surfaces, yet below the horizon.

  Mon 12v08 We discussed the Scott and Polvani shallow-water-model on a sphere and more thoroughly Lee's two-level model on a sphere driven by a meridional temperature gradient. The idea of spontaneously appearing jets is that the beta-effect organizes energy and has a distinct preference for zonal flows. Not unlike the way a stratified, non-rotating fluid will respond to an episode of stirring: First there is some isotropic turbulence, but then the stratification squeezes the mixed fluid out horizontally. As the energy disperses and dies away the motions have a lower natural frequency (estimated by the W/L 'overturning' frequency, where L is an eddy diameter and W a typical vertical velocity). This lowering natural frequency of the eddies mates well with internal gravity waves that propagate with group velocity ever more horizontal at lower frequency.

  It seems natural that free flow can occur in the horizontal but not the vertical with stratification, and analogously, free flow on a rotating Earth can occur in the zonal velocity but not the meridional velocity. 'PV elasticity' opposes meridional velocities, turning them into Rossby waves, and this elastiticity is very scale dependent: it works for large L scales but not for small L scales (use the Rossby wave frequency, ~ beta*L, as a guide to the 'elastic' restoring force). Small scale turbulence develops an 'upscale cascade', that is the eddies cluster together and make larger L scales, and this makes them more susceptible to the PV elasticity as beta*L increases.

  We spoke of the stratified dynamics problem which is central to jet streams. Just as we have seen momentum transport by Rossby waves in the x-y plane we now see it in the x-z plane, possible also associated with upward propagating baroclinic Rossby waves. In both cases the Eulerian thinking ( averaged along a fixed latitude circle) can be understood through the pressure force exerted across a wavy rigid boundary. This was a wavy shaped sidewall in the x-y Rossby wave problem but then becomes a wavy mountain range in the x-z upward radiation problem.

  The final step into the full baroclinic problem is to realize that marked fluid surfaces...marked with dye, or in the x-z problem they may be potential temperature surfaces...transmit horizontal momentum the same way that rigid wavy boundaries do: through pressure forces correlated with the slope of the surface.

  This idea helps to understand the two parts of the EP flux, (or equivalently the two parts of the PV flux, which equals d/dy of the EP flux). The new effect, the vertical transport of horizontal momentum by pressure forces along wavy theta-surfaces, appears in EP as the vertical derivative of a meridional eddy heat flux. The connection between the two is through the relation between a theta perturbation, theta', and a vertical displacement of a theta-surface, z' = theta' = z' d(theta bar)/dz Thus {v'theta'} is proporational to {v'z'}, the correlation of meridional velocity and z-coordinate of the surface. For quasi-geostrophic flow v' is proportional to dp'/dx, so this is {v'z'} ~ {dp'/dx z'} ~ -{p' dz'/dx} or the correlation of pressure and slope, which is the horizontal pressure force. Above I use { } brackets for zonal averaging because HTML uses < > for other purposes!
  
  We meet Monday 12 May at 11.00 in 319 Ocean Sciences (my office) papers to read (Rahul will lead discussion):
  Scott & Polavani jets (1 layer) JAS 07
  Lee jets (2 layer) JAS 05
  
  

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  Week 6. 5v-9v08
  

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  Week 5. 28iv-2v08
  
  Linear theory for stationary Rossby waves in uniform westerly flow over a cylindrical mountain (McCartney, 1976).
  The notes on Rossby wave and mean flow generation, in new and improved form (with some sketches and additions and corrections in blue) are here.

  We went over Randall and Held JAS 91 led by Chaim, and compared the meridional flux of zonal momentum there with various atlas plots..showing particularly the 2-lobed structure in the S Hemisphere, with more like one lobe in NH. Two lobes that is, positive south of the wave source and negative north of the wave source is what we expect from simple wave theory in a channel model. That is, the herring-bone pattern. Libby's plots showed how strikingly this varies with season. Generally we argued that poleward of the jet there is relatively small space, so that the momentum flux could saturate the way it does when a rigid wall is placed poleward of the wave source (the reflected waves cancel the incident waves in their ). On a sphere Rossby wave refraction by variation of beta with latitude (the waves ideally follow great circle paths) tends to favor Equatorward propagation, though greatly affected by zonal flow variations.

  We looked at barotropic numerical experiments in which Rossby waves were generated at the center of a beta-plane channel. Banana-shaped wave crests develop with clear tilts. A mean westerly jet grows at the latitudes spanned by the forcing (think of it as a circular disk oscillating up and down, at a single frequency, in the center of the channel). The u'v' tilts depended on longitude, the characteristic two lobes (as a function of y) spreading out as one goes west of the forcing region. The mean flow is generated west of the forcing (remember that there is no pre-existing zonal flow here). It is in fact a low-frequency Rossby wave that takes the form of a nearly zonal wind.

  Assignment:
  o read Randall & Held handout Chaim to lead the discussion on this?
  o read Vallis:
  295-301 (7.1-7.2.2) EP flux
  485-503 (12.1-12.2.1) single layer MOM analysis
  with these as background:
  229-232 (5.7.1) single layer Rossby wave
  236-241 (5A) group velocity
  o look at meridional flux of zonal momentum in your reanalysis dataset (you can also look in the atlases like ERA40 but it's interesting exploring with 'new' data). What are the space and time structures that are evident?**
  o reread (re-read?) my Rossby wave generation problem from this week.
  

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  Week 4. 21-25iv08
  

  Our discussion of Branstator JClim 2002 (the circumpolar waveguide) was very interesting. He looks at low-frequency variability, either winter season averaged winds or monthly averaged winds. The covariance maps show striking ducting along the stationary wave/jet stream pathway. The scales are wavenumber 5-ish (the 5 pointed star). The fact that this is low frequency and not transient eddy (synoptic storms) energy is important, yet the intrinsic dynamics of low-freq and high freq waves can be similar (both might be simple Rossby waves riding on different mean zonal winds). Hoskins & Ambrizzi's barotropic study, sending waves along the wintertime mean westerly wind field, shows also low-freq response (because the forcing is steady) yet the group velocity causes the standing pattern to set up over just a few days. Connecting this with high-freq transient eddies/storm tracks will be very interesting!
  
  
  

  Held et al, Branstator and Hoskins et al all note that a snapshot of the westerly winds shows narrow jetstreams, and so the background wintertime mean flow is never there by itself. This suggests another approach to modeling the jet stream wave guide, perhaps as a sharp front of PV which has its own intrinsic wave propagation (see detail in Held et al. J Clim 2002 paper on stationary waves).

  We need to augment these ideas with basic ray theory for Rossby waves propagating in a zonal flow U(y). The tropical critical line where U(y)=c, the westward phase speed, is crucial. Stationary waves have c=0, so their critical line is the transition latitude where westerlies become easterlies. Hoskins & Ambrizzi point out breaks in the jet stream waveguide in the eastern Pacific and elsewhere, suggesting that the waveguide is less than circumpolar. Their diagram of the broken waveguide is just below:
  
  

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  Week 3. 14-18iv08
  

  We looked at Rahul's plots of downstream development in an idealized channel flow, with a developing baroclinic wave. Perturbations were placed both on initially simple zonal jet and on a fully turbulent, fully developed jet. Libby's plots of jet stream zonal velocity showed an interesting summer max north of Tibet; this came out in Brian Hoskin's great talk on monsoons and summer subtropical anticyclones; in summer the jet moves north, and is in part responding to blocking by the Himalayan Plateau (rather than textbook flow up and over it). Chaim showed some energy maps for high and low AO index, bringing out the tropopause level KE which far exceeds the stratospheric KE despite the higher velocity aloft. PBR showed some plots of the mass transport in the westerly winds. This averages about 550 Sverdrups in winter (between 40N and 60N). 1 Sverdrup=10⁹ kg/sec. Major ocean currents have up to 200 Sverdrups of transport (Antarctic Circumpolar Current). The Gulf Stream ranges from 30 to 120 Sverdrups or so. The meridional circulations are weaker, though Oort & Peixoto's plot suggested ~ 200 Sverdrups in the Hadley cell and only 2 to 4 Sverdups in the Ferrel cells. Oceanic MOC's amount to about 16 Sverdrups in the Atlantic. Note that a mere 1 Sverdrup of water vapor is carried poleward by the atmospheric meridional overturning (the specific humidity being very small on average). This water vapor represents 2 petawatts (10¹⁵ watts), fully 1/2 of the total poleward energy flux.

  The Hadley cell discussion emphasized the simple idea that its meridional width is an appropriate Rossby radius. With a limited store of APE (measured by the pole-to-Eq temperature difference) only so much KE can arise, and with angular momentum conserving flow that greatly restricts the distance fluid rings can travel symmetrically poleward.

  At the poleward end of the Hadley cell the angular momentum conserving westerlies form a nice jet…our first real jet stream. The slumping horizontal stratification and its geostrophic adjustment was carried out on the board. This is a sort of piece of the symmetric Hadley cell, in which the fluid generates its own vertical stratification N, and in doing so moves horizontally a distance equal to the Rossby radius NH/f. Generally the Rossby radius = Co/f where Co is the phase speed of long, hydrostatic gravity waves in the vertical mode appropriate to the problem (here, internal gravity waves). This problem can, like Held & Hou's Hadley cell, be relaxed back to an equilibrium temperature field and thus made steady with dissipation.

  We are beginning to look at lateral momentum flux in 1-layer fluids.

  assignment:

  • Read Branstator, G., Circumpolar teleconnections, the jet stream waveguide and the N Atlantic Oscillation, J Climate 2002
    
  • Think about how to do some analysis of your one-year reanalysis dataset for:
    momentum flux (using EP flux is one way..but this has to be zonally averaged; there are ways to generalize this to 3D), and for energetics…just APE and KE right now.
    
  • select a term project

  Below are plots of (1) artificially generated maps of pressure, u and v velocities and vertical vorticity ζ for 3 choices of wavenumber spectra shown. Note how the distinction between p, u,v and ζ depends on the breadth of the spectrum. Below that (2) are a dynamical pair, pressure or streamfunction and ζ. These are much more 'fluid' than the artificial fields in (1). Think about how plots of θ would look; it's quite subtle.
  
  left: streamfunction/pressure.................... right: ζ
  

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  Week 2. 7-11iv08
  Missed Tuesday because of trip to snowy Faroes Islands with Al Gore (sort of) for a climate meeting. Thursday discussed more of JM Wallace stationary waves paper (83) and a few basic ideas relating to Held et al 2002 stationary waves paper. Chaim led us to talk about Eq'ward propagation in the EP flux field for stationary waves...looking at waves 1,2 with high altitude Eqward vectors and waves 5,6 with tropopause level prop. Strong need to see how close these are to linear Rossby wave propagation and in which vertical mode. Basic shallow water model with just the barotopic mode simulating the external mode of the stratified system...shows the ray paths for Rossby waves on a sphere bending Eqward, then back and forth across the Equator...but following great circle paths on the true sphere. The turning latitude, that is the highest latitude they can reach, is due to the smallness of beta up there. With beta varying in latitude, look at separable solutions psi = g(y)exp(ikx-i omega t). Ray theory (short wavelength compared to radius of the Earth) shows refracting ray paths, and on the wavenumber plane (k,l), the circles of constant omega for various latitudes help us to construct the ray paths.

  EP flux or Reynolds stress is visible in these Rossby wave wavecrest patterns, just as with the legendary 'tilted trough' in the westerly winds. Poleward group velocity >0 where <0 and conversely. This sets the stage for the sharpening of broad westerly winds into jet streams, insofar as a shallow water model can do such things. We will begin some of the Vallis Ch. 12 material on this next week.

  We talked about where the highs and lows are situated in flow over major orographic features...showing how the shallow-water f-plane result (H over the mountaintop due to vortex squashing) changes on the beta-plane. Slow flow follows PV contours which here are f/h = constant curves where f = fo + beta*y. These dip Eq'ward over a ridge or mountain, putting a cyclonic L over the mountain! But then we speed up the flow and it blows the L center downwind. The L thus sits over the lee slope and creates wave drag...pressure force p dh/dx on the mountain. THe equal and opposite force on the atmosphere is carried by Rossby wave EP flux far away from the mountain. And, we shall see, this makes jets.

  This just in...Elizabeth sent me this url for Prof. Wallace's lectures on general circulation this term. They are very worth looking at: www.atmos.washington.edu/2008Q2/545/

  assignment: for Tues 15iv2008
  (i) do your taxes
  (ii) look deeper into Vallis Ch. 11 on the Hadley cell. It is pretty tough going because the assumptions are not always clear (for example the assumption that the zonal velocity goes toward zero at the ground). The original paper is Held & Hou JAS 1980 which may help you. Note how a subtropical jet does form at the poleward end of the Hadley cell in this model.
  (iii) see if you make sense of my handout page on geostrophic adjustment, which has a very similar aim as (ii) in a much simpler context. We take an initial horizontally stratified fluid and let the fluid 'slump' under gravity, then see the horizontal adjustment arrested by rotation..a thermal wind field is set up that holds the 'polar front' in place. The Rossby radius emerges naturally as the distance slumped, and the fluid develops its own vertical density stratification and zonal wind where there was none at the beginning. If it's not clear I will explain the notes on Tuesday.

  We can actually try to change this initial-value problem into a steady circulation rather like the Hadley cell model by forcing the temperature equation toward a horizontally stratified reference temperature, similar to Held&Hou. Anyone like to try?
  (iv) If you have time start looking at Vallis Ch 12.1 on what maintains the westerly jet (in a shallow-water one-layer model).
  Have a think about your term project. These 10-week terms race by quickly. Grown-up universities have semesters rather than quarters.
  and finally
  big (v) Do some more looking at the jet stream with reanalysis data. We will make this the FIRST item on Tuesday.

  Ideas:

  • videos of zonal velocity (merid sections at particular longitudes).
  • what is the mass transport in the jet stream/westerlies in megatonnes per sec? We oceanographers talk about Sverdrups which are units of 10⁹ kg/sec (megatonnes), and our major currents have transports of 50 to 200 Sverdrups. How does the atmosphere compare? While you are at it, use Matlab to tell you about the meridional mass transport across various latitude circles. Can you see the shape of the Hadley, Ferrel and polar overturning suggested by this merid transport?
  • can you figure how the high latitude jet behaves near the Himalayan plateau? How does it round the topography, with what sort of vertical structure? What do you see in the downwind wake, over China? How does that wake behave when it comes out over the Pacific?
  • how much of the KE of the northern hemisphere westerlies is associated with jet stream like concentrations of the winds? What is the vertical structure of this KE, for example we know there is a lot at the tropopause, but how much in the polar vortex in the stratosphere overhead?
  
  

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  Week 1.
  Eventually you will see slides from class here and will link your webpages too. I'll try to summarize any blackboard work we do, for example velocity spirals (backing and veering, or 'thermal wind 102').
  
  assignment:
  • read JM Wallace stationary wave paper from Hoskins & Pearce volume Large-scale dynamical processes in the atmosphere, Academic Press, 1983 (handed out on paper) and now read the more advanced dynamical model discussion of stationary waves by Held, Ting and Wang, J.Climate 2002. This is a pretty intricate paper describing the effects of orography (mountains) and land/sea thermal forcing of these long waves...but have a go at it!
    Chaim will lead discussion on Thurs 10 iv 2008
  • take an individual 1 year of data from NCEP or ECMWF/ERA40 pressure level data at 17 levels (in the NCEP case) and make plots illuminating the jet streams:
    where intense? seasons? levels? meridional structure? map on p levels. What defines best the jet streams? How do N and S hemispheres differ? Using kinetic energy as a gauge of amplitude you overlay different fields (for example theta and zonal velocity in meridional sections). Relation with orography and land/ocean domains?
    While it's extra work, begin to think about mapping on a new vertical coordinate like theta or even PV.

    It would be nice if there were a smart electronic atlas for atmospheric reanalysis data. Anyone know of one? We have produced these for ocean data, and it is good to browse at will through the 4 dimensions. As it is you can visit Climate Explorer to regress almost any famous time series (eg. NAO) with almost any field ( climexp.knmi.nl) and of course the atmos.washington.edu site has great plots and animation potential for set levels and domains.

  • Read Vallis early part of Ch.11 on the Hadley cell. Try to give a simple scale analysis that predicts the meridional width of the Hadley cell, given the amplitude of the potential temperature difference between the thermal Equator and its poleward limit. The argument is similar for the figure below, from the GFD lab:
    
    
    
    The orange fluid is injected quickly (all at once) from the vertical pipe, with density intermediate between the two densities of the layers. Its outward motion, conserving absolute angular momentum, develops an anticyclonic spin. The outward motion is then 'arrested'..stopped as it comes into thermal wind balance. With this azimuthal spin dying off both above and below, a thermal wind tilt of the density interfaces, as seen here, can do this. A slight trace of purple dye shows the shear lines above and below the vortex. Think about what determines the diameter of the vortex, given that we know the density difference, rotation rate, layer thickness. Roughly how would you expect the velocity to vary with radius from the center?
    The 2d image shows the vortex after it has gone unstable and turned into a spinning 'spiral nebula'.
    The 3d image shows in purple permanganate streaks a vertical profile of the azimuthal (round-and-round) component of velocity. THis is the basic thermal-wind shear of the vortex. It is NOT the radial velocity as it might appear.