The Vorticity Equation in a Rotating Stratified Fluid

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1 Capter 7 Te Vorticity Equation in a Rotating Stratified Fluid Te vorticity equation for a rotating, stratified, viscous fluid Te vorticity equation in one form or anoter and its interpretation provide a key to understanding a wide range of atmosperic and oceanic flows. Te full Navier-Stokes' equation in a rotating frame is D u + f u = pt g k + ν u were p is te total pressure and f = fk. We allow for a spatial variation of f for applications to flow on a beta plane.

2 D u + f u = pt g k + ν u Now u u = ( u ) + ω u d i b g + u + ω + f u =- p k + ν u t T g take te curl or D ω + f = ω + f u ω + f u+ p T + ν ω Dω = u f +... Note tat [ ω + f] u] = u (ω + f) + (ω + f) u - (ω + f) u, and [ω + f] 0. Terminology ω a = ω + f is called te absolute vorticity - it is te vorticity derived in an a inertial frame ω is called te relative vorticity, and f is called te planetary-, or background vorticity Recall tat solid body rotation corresponds wit a vorticity Ω.

3 Interpretation D ω + f = ω + f u ω + f u+ p T + ν ω Dω is te rate-of-cange of te relative vorticity u f: If f varies spatially (i.e., wit latitude) tere will be a cange in ω as fluid parcels are advected to regions of different f. Note tat it is really ω + f wose total rate-of-cange is determined. D ω + f = ω + f u ω + f u+ p T + ν ω aω + ff u consider first ω u, or better still, (ω/ ω ) u. unit vector along te vortex line ω u = = ( ω ) + ( n + b ) s s u s s u n u b u n n + u b s δs u s ω u u + δu principal normal and binormal directions te rate of relative vorticity production due to te stretcing of relative vorticity te rate of production due to te bending (tilting, twisting, reorientation, etc.) of relative vorticity

4 D ω + f = ω + f u ω + f u+ p T + ν ω f u = f = f + f k te rate of vorticity production due to te bending of planetary vorticity a f ω + f u te rate of vorticity production due to te stretcing of planetary vorticity = (/)(D/)(ω + f) using te full continuity equation a relative increase in density a relative increase in absolute vorticity. Note tat tis term involves te total divergence, not just te orizontal divergence, and it is exactly zero in te Boussinesq approximation. D ω + f = ω + f u ω + f u+ p T + ν ω p T sometimes denoted by B, tis is te baroclinicity vector and represents baroclinic effects. B is identically zero wen te isoteric (constant density) and isobaric surfaces coincide. Denote φ = ln θ = s/c p, s = specific entropy = τ ln p T ln + constants, were = τ = κ. p T B = φ

5 B = φ p T B represents an anticyclonic vorticity tendency in wic te isentropic surface (constant s, φ, θ ) tends to rotate to become parallel wit te isobaric surface. Motion can arise troug orizontal variations in temperature even toug te fluid is not buoyant (in te sense tat a vertical displacement results in restoring forces); e.g. frontal zones, sea breezes. ν ω represents te viscous diffusion of vorticity into a moving fluid element. Te vorticity equation for synoptic scale atmosperic motions Te equations appropriate for suc motions are t + u u + w + f u = p (a) and p 0 = + σ Let v v ω = u =,, x y Take te curl of (a)

6 ω + ( ω + f) u + u ( ω + f) t ω ( ω + f) u + w + w = + p d i a f u u = We use u + ω u and φa = φ a+ φ a Te vertical component of tis equation is ζ ζ v = u ( ζ+ f) w ( ζ+ f) + + t x y v p p + y x x y y x were ζ = k ω = v / x / y An alternative form is D v ( ζ+ f) = ( ζ+ f) + + " + " x y Te rate of cange of te vertical component of absolute vorticity (wic we sall frequently call just te absolute vorticity) following a fluid parcel. Te term ζ+ ( f) + x v is te divergence term y For a Boussinesq fluid: / x + v/ y + /= 0 => v ζ+ ( f) + = ( ζ+ f) x y

7 v ζ+ ( f) + = ( ζ+ f) x y ζ + f w + dw w (ζ + f)/ corresponds wit a rate of production of absolute vorticity by stretcing. For an anelastic fluid (one in wic density variations wit eigt are important) te continuity equation is: / x + v/ y + (/ 0 ) ( 0 w)/ = 0 => v ( 0w) ζ+ ( f) + = ( ζ+ f) x y 0 v Te term in te vorticity equation is te y x tilting term; tis represents te rate of generation of absolute vorticity by te tilting of orizontally oriented vorticity ω = ( v/, /, 0) into te vertical by a non-uniform field of vertical motion (/ x, / y, 0) 0. w(x) ξ = v

8 Te last term in te vorticity equation is te solenoidal term. Tis, togeter wit te previous term, is generally small in synoptic scale atmosperic motions as te following scale estimates sow: L NM L NM v W U y x QP H L = 0 O O p p δ δp x y y xqp L = 0 s, s Te sign indicates tat tese may be overestimated due to cancellation. ; End of Capter 7