Geopotential tendency and vertical motion

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Vernon Jones
6 years ago
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1 Geopotential tendency and vertical motion

2 Recall PV inversion Knowin the PV, we can estimate everythin else! (Temperature, wind, eopotential ) In QG, since the flow is eostrophic, we can obtain the wind field exactly. Then with thermal wind balance, obtain the temperature exactly. More enerally (i.e., not QG), need to assume a number of balance conditions to do the PV inversion. Example: PV inversion with ozone Ozone concentrated in stratosphere PV concentrated in stratosphere (stroner stability) PV conservation means PV behaves like a passive tracer (much like ozone) So, could use measurements of ozone to uess PV field then invert this derived PV to obtain wind field Then use thermal wind equation to deduce temperature field Would be useful near tropopause and lower stratosphere where wind measurements are difficult Would this work?

3 Ozone measured from TOMS PV calculated from weather observations Davis et al. (1999), QJRMS, 15, hpa wind from PV ozone Davis et al. (1999), QJRMS, 15,

(1985), QJRMS, 111, 877-946 Example: Vertical couplin forced at 5 hpa Amount of vertical inversely proportional to lenth

4 Example: more PV anomalies PV near tropopause PV near surface - P > L P < - P > L - H PV anomaly shaded Potential temperature, and wind anomalies contoured - P < H Hoskins et al. (1985), QJRMS, 111, Example: Vertical couplin forced at 5 hpa Amount of vertical inversely proportional to lenth scale (as Laplacian selects for smaller scale). Consider scale (via m -1 ): ψ = sin mz ψ = mcos mz z ψ = m sin mz z = m ψ lare m small m Aain, an artifact of the smoothin associated with the eleptic

5 Example: Impact of aspect ratio Recall vertical and horizontal scales are related to f and σ. Specifically, k/m ~ f /σ Consider scale (via k -1 ): ψ = sin mz ~ sin kx ψ ~ k cos kx z ψ ~ k sin kx z ~ k ψ So more vertical propaation for loner waves (k small, k -1 lare) lare k small k In future we will see that this means, for lon waves, upper troposphere flow, can mess with lower troposphere flow.

6 Geopotential tendency (a synoptic meteoroloy view of QG) i.e., PV is nice and all, but eopotential makes more sense to me! Pronostic eopotential equation QG vorticity equation: QG thermodynamic equation: ζ = V V ( ζ f ) f Φ σω = ω R c p J p As a consequence of conservation of QG PV f Φ Φ f = f Φ V f V σ f σ 1 Notice left hand side is an elliptic operator for the tendency (which can be solved iven boundary conditions) Terms on the riht are all in terms of eopotential

7 Geopotential tendency equation f Φ = f Φ V f V σ f 1 vorticity advection f Φ σ temperature advection Thermal advection dominates development. (stroner in lower troposphere) Throuh thermal wind balance, ensures that low level development is associated with upper level chane (surface cycloenesis causes upper level trouh, etc) Notice this suests a transfer of enery from potential enery (related to thickness/oepotential heiht) to kinetic enery (more vorticity, and stroner wind speeds) This conversion is fundamental to the eneral circulation ζ QG vorticity 1 = Φ f QG prediction ζ = V ( ζ f ) Given Φ, can compute u and v. What about vertical velocity? 1 vorticity advection w f vorticity stretchin (aeostrophic) Eliminate it! (Use thermodynamic equation.) f Φ Φ f = f Φ V f V σ f σ vorticity Vertical difference advection in temperature advection Recall Laplacian ives dφ < when dζ >, etc Invertin Laplacian tends to smooth. (And ive a minus sin) Also, elliptic equations have solutions!

Geopotential tendency f Φ Φ f = f Φ V f V σ f σ 1 Φ falls with (positive) vorticity advection (cyclonic) Φ falls with when warm air advection increases with heiht (or when cold air advection

8 Geopotential tendency f Φ Φ f = f Φ V f V σ f σ 1 Φ falls with (positive) vorticity advection (cyclonic) Φ falls with when warm air advection increases with heiht (or when cold air advection decreases with heiht) Example: Warm air advection at the surface causes increased thickness Increased thickness causes hih pressure at layer top Hih pressure creates aeostrophic horizontal motion Unbalanced aeostrophic drives diverence (mass lost from layer) Diverence lowers heihts, and creates low pressure at surface (still hih pressure at layer top ) Surface low causes converent aeostrophic motion for balance So knowin eopotential can estimate "omea" with QG!

9 QG vertical motion ω was eliminated in prediction equation, but is important for, for instance, stretchin. Specifically, if we want the aeostrophic diverence, need to be able to extract it from horizontal eostrophic flow Different estimates of ω are subject to different errors in equations (this is a practical issue) Consider momentum equation? Continuity? Thermodynamic? Each ω as a residual from a difference in lare quantities. ua x va y ω = V Φ σω = R c p J p Expect most accurate method to be based on state of eopotential rather than chane in eopotential (as, e.., in vorticity equation)

Quasi-geostrophic system

Quasi-geostrophic system Quasi-eostrophic system (or, why we love elliptic equations for QGPV) Charney s QG the motion of lare-scale atmospheric disturbances is overned by Laws of conservation of potential temperature and potential