Diagnosing Vertical Velocity: The Omega Equation. Robert Todd and Kaushik Srinivasan 15 May 2009

Transcription

1 Diagnosing Vertical Velocity: The Omega Equation Robert Todd and Kaushik Srinivasan 15 May 2009

2 Introduction The vertical velocity is much smaller then the horizontal velocity and therefore, difficult to measure. A possible approach - measure all other fields (density, horizontal velocity) and use relevant dynamical equations to infer the vertical velocity. Why do we care? - Upwelling advects nutrients and particulate matter from the bottom layer to the euphotic zone.

3 Quasigeostrophic Equations Governing Equations: D g u g Dt D g v g Dt fv a =0 + fu a =0 Geostrophy: fu g = 1 ρ 0 ( p x, p y ) D g ρ Dt + w d ρ z dz =0 D g Dt ( ρg )+wn 2 =0 ρ 0 Thermal Wind: u g z = g ρ fρ 0 y v g z = g fρ 0 ρ x

4 Deriving the Omega Equation Vertical derivative of equations of motion and horizontal derivatives of density equations: D g Dt u gz = Q x + fv az D g Dt v gz = Q y fu az g D g ρ 0 Dt ρ x = Q x +(N 2 w) x g ρ 0 D g Dt ρ y = Q y +(N 2 w) y

5 Reprise: The frontogenesis vector Q

6 Reprise: The frontogenesis vector Q Consider Yvonne's case, horizontal uniform strain field so,

7 The QG Omega Equation Subtracting: x (N 2 w) f 2 u a z =2Q x y (N 2 w) f 2 v a z =2Q y From continuity: w z = ( ua x + v a y ) Gives us the QG Omega equation: ( 2 x y 2 )(N 2 w)+f 2 2 w z 2 =2 Q

8 A more physical interpretation of Q Suppose we neglect the horizontal derivatives This works if L h >L R f 2 2 w z 2 2 Q Assume(hope?) that most of the energy is in the baroclinic mode. (Not OK for Fronts!) Then w is like a sine function -> w and 2 w z 2 have opposite signs Then w and 2 w z 2 have opposite signs w and i.e. w has the same sign as Q Look at maps of Q and figure out upwelling and downwelling

9 Application of Omega Equation { 2 (N 2 w)+f 2 2 w ( z =2 Q 2 ) Q = g ug ρ 0 x ρ, u g y ρ Need (1) statically stable density field and (2) absolute geostrophic velocities. Measurements: Density from SeaSoar (~3.5 km across front, ~25 km along front) Absolute horizontal velocity from ADCP Procedure: 1. Objectively map in horizontal levels 2. Adjust mapped fields 3. Solve Omega Equation

10 Application of Omega Equation Inferring geostrophic velocity: (u g,v g ) = R = ( ψ y + R y, ψ x R x g z ρdz fρ 0 z 0 ), Goal: Find streamfunction for geostrophic velocity as close as possible to mapped (u,v) V w u [(u g u) 2 +(v g v) 2 ]dv Minimize w.r.t. ψ and solve with appropriate B.C. s Result: Geostrophic velocity very similar to (u,v). RMS difference: cm s -1 Complex correlation: Rudnick 1996, JGR

11 Application of Omega Equation Solving the Omega Equation: { 2 (N 2 w)+f 2 2 w ( z =2 Q 2 ) Q = g ug ρ 0 x ρ, u g y ρ Boundary Conditions: Surface: w = 0 Bottom: w = 0 at z = 2524 m Lateral: w = 0 well outside observation region Forcing: Must specify forcing outside of observation region Q 0 from chowder.ucsd.edu after Rudnick 1996, JGR Result: Maximum vertical velocities on the order of 2 x 10-4 m s -1 (~20 m d -1 )

12 Vertical Heat Flux Method: Estimate heat flux from covariance of density and vertical velocity ρ w Assume no salinity variability Average over survey region Result: Positive flux from rising water on light side and from sinking water on dense side Maximum flux at m, strongest in March surveys Result is heating of upper layer, cooling of lower later ==> Restratification! Rudnick 1996, JGR

13 Vertical Ageostrophic Circulation Cells of ageostrophic flow with upward flow on light side of front feeding downward flow on the dense side [Pollard and Regier 1992, JPO] [Rudnick 1996, JGR]

14 Recent Developments Other formulations of the Omega equation [Pallàs Sanz & Viúdez 2005]: Quasigeostrophic and semigeostrophic Omega equations underestimate (overestimate) vertical velocity when potential vorticity is negative (positive) Inviscid, isentropic, Boussinesq, f-plane version of Omega equation (approximate): N 2 2 hw + f(f + ζ)w zz = 2 h Q h + fζ ph 2 hu h, Q h h u h h ϱ Large ageostrophic flow in survey area lead to significant differences between vertical estimated vertical velocities, but similar qualitative structure [Pallàs Sanz & Viúdez 2005, JPO]

15 Recent Developments Lack of synopticity [Rixen, et al. 2001]: Omega equation needs synoptic fields, but surveys not synoptic Attempt to correct location of measurement using horizontal geostrophic currents Effective?? [Rixen, et al. 2001, DSRI]