Diagnosing Vertical Velocity: The Omega Equation Robert Todd and Kaushik Srinivasan 15 May 2009
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.
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
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
Reprise: The frontogenesis vector Q
Reprise: The frontogenesis vector Q Consider Yvonne's case, horizontal uniform strain field so,
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 2 + 2 y 2 )(N 2 w)+f 2 2 w z 2 =2 Q
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
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
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: 0.06-0.07 cm s -1 Complex correlation: 0.97-0.98 Rudnick 1996, JGR
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 )
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 100-150 m, strongest in March surveys Result is heating of upper layer, cooling of lower later ==> Restratification! Rudnick 1996, JGR
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]
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]
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]