Chapter 7

The Vorticity Equation in a Rotating Stratified Fluid

The vorticity equation for a rotating, stratified, viscous fluid

» The vorticity equation in one form or another and its interpretation provide a key to understanding a wide range of atmospheric and oceanic flows. » The full Navier-Stokes' equation in a rotating frame is

Du 1 +∧fu =−∇pg − k +ν∇2 u Dt ρ T where p is the total pressure and f = fk.

» We allow for a spatial variation of f for applications to flow on a beta plane. Du 1 +∧fu =−∇pg − k +ν∇2 u Dt ρ T

1 2 Now uu=(u⋅∇ ∇2 ) +ω ∧ u

∂u 1 2 1 2 +∇2 ufuku +bω +g ∧=- ∇pgT − +ν∇ ∂ρt di

take the curl

D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2 Dω or =−uf ⋅∇ +.... Dt

Note that ∧ [ ω + f] ∧ u] = u ⋅ (ω + f) + (ω + f) ⋅ u - (ω + f) ⋅ u, and ⋅ [ω + f] ≡ 0.

Terminology

ωa = ω + f is called the absolute vorticity - it is the vorticity derived in an a inertial frame

ω is called the relative vorticity, and

f is called the planetary-, or background vorticity

Recall that solid body rotation corresponds with a vorticity 2Ω. Interpretation

D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2

Dω is the rate-of-change of the relative vorticity Dt

−⋅∇uf: If f varies spatially (i.e., with latitude) there will be a change in ω as fluid parcels are advected to regions of different f.

Note that it is really ω + f whose total rate-of-change is determined.

D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2

afω +⋅∇fuconsider first ω ⋅ u, or better still, (ω/|ω|) ⋅ u. unit vector along the vortex line ∂u ∂ ∂ ω= ⋅∇u = (uuu ω++ )(nb  ) ∂ss∂ snb∂ s

principal normal and binormal δs directions u + δu u ω s the rate of relative vorticity production due to the stretching of relative vorticity u uunb +  the rate of production due to the bending (tilting, ns twisting, reorientation, etc.) of relative vorticity D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2

∂uu∂ ∂w fu⋅∇ =f =f h +f k ∂z ∂z ∂z the rate of vorticity production due to the bending of planetary vorticity

the rate of vorticity production due to −+∇⋅aω fuf the stretching of planetary vorticity

= (1/ρ)(Dρ/Dt)(ω + f) using the full continuity equation

a relative increase in density ⇒ a relative increase in absolute vorticity.

Note that this term involves the total divergence, not just the horizontal divergence, and it is exactly zero in the Boussinesq approximation.

D 1 2 afafafωω+=ffufu +⋅∇−+∇⋅+∇ρ∧∇+ ωpT ν∇ ω Dt ρ2

1 ∇ρ ∧ ∇p sometimes denoted by B, this is the baroclinicity vector ρ2 T and represents baroclinic effects.

B is identically zero when the isoteric (constant density) and isobaric surfaces coincide.

Denote φ = ln θ = s/cp, −1 s = specific entropy = τ ln pT − ln ρ + constants, where = τ−1 = 1 −κ. 1 B = ρ ∇pT ∧ ∇φ 1 B = ρ ∇pT ∧ ∇φ

» B represents an anticyclonic vorticity tendency in which the isentropic surface (constant s, φ, θ ) tends to rotate to become parallel with the isobaric surface.

» Motion can arise through horizontal variations in temperature even though the fluid is not buoyant (in the sense that a vertical displacement results in restoring forces); e.g. frontal zones, sea breezes.

ν∇2ω represents the viscous diffusion of vorticity into a moving fluid element.

The vorticity equation for synoptic scale atmospheric motions

The equations appropriate for such motions are

∂u ∂u 1 h +⋅∇+uuwph +∧=−∇fu (a) ∂tzh hh∂ρh

1 ∂p and 0 =− +σ ρ ∂z ⎛⎞∂vuv∂∂ ∂ u Let ω=∇⋅hhu =−⎜⎟,, − ⎝⎠∂zzx∂∂ ∂ y Take the curl of (a) ∂ω h ++∇⋅+⋅∇+()ωωfuu () f ∂t hhhhh ∂ω ∂u 1 −+⋅∇+()ω fuwwhh +∇∧=+∇ρ∧∇ p h h ∂zz∂ρ2 h

1 2 We use uuhh⋅∇ =∇di2 u h +ωφφ∇ hh ∧ uand ∇∧a af =∇φ∧ a + ∧ a

The vertical component of this equation is

∂ζ ∂ζ⎛⎞ ∂uv ∂ =−uh ⋅∇(f)w ζ+ − − (f) ζ+⎜⎟ + + ∂∂∂∂tzxy⎝⎠ ⎛⎞⎛⎞∂wu∂∂∂ wv 1 ∂ρ∂∂ρ∂ p p ⎜⎟⎜⎟−+2 − ⎝⎠⎝⎠∂yz∂∂∂ρ∂∂∂∂ xz xy yx where ζ = k ⋅ ω h = ∂vx//∂ − ∂ uy∂

An alternative form is

Duv⎛⎞∂∂ ⎛⎞⎛⎞ (f)(f)ζ+ =− ζ+⎜⎟ + +⎜⎟⎜⎟ " + " Dt⎝⎠∂∂ x y ⎝⎠⎝⎠

» The rate of change of the vertical component of absolute vorticity (which we shall frequently call just the absolute vorticity) following a fluid parcel.

⎛⎞∂uv∂ The term −ζ+ (f) ⎜⎟ + is the divergence term ⎝⎠∂xy∂ For a Boussinesq fluid: ∂u/∂x + ∂v/∂y + ∂w/∂z= 0 => ⎛⎞∂uv∂∂ w −ζ+(f)⎜⎟ + = (f) ζ+ ⎝⎠∂xy∂∂ z ⎛⎞∂∂uv ∂ w w + dw −ζ+(f)⎜⎟ + = (f) ζ+ ⎝⎠∂∂xy ∂ z ζ + f w

(ζ + f)∂w/∂z corresponds with a rate of production of absolute vorticity by stretching.

For an anelastic fluid (one in which density variations with height are important) the continuity equation is:

∂u/∂x + ∂v/∂y + (1/ρ0)∂(ρ0 w)/∂z= 0 =>

⎛⎞∂∂uv 1∂ρ(w)0 −ζ+(f)⎜⎟ + = (f) ζ+ ⎝⎠∂∂xy ρ∂0 z

⎛⎞∂∂wu ∂∂ wv The term⎜⎟− in the vorticity equation is the ⎝⎠∂∂yz ∂∂ xz tilting term; this represents the rate of generation of absolute vorticity by the tilting of horizontally oriented vorticity

ωh = (−∂v/∂z, ∂u/∂z, 0) into the vertical by a non-uniform field of vertical motion (∂w/∂x, ∂w/∂y, 0) ≠ 0.

∂v w(x) ξ =− ∂z » The last term in the vorticity equation is the solenoidal term. » This, together with the previous term, is generally small in synoptic scale atmospheric motions as the following scale estimates show:

∂w ∂u ∂w ∂v W U L − O ≤=10−−11s 2 , NM ∂y ∂z ∂x ∂zQP H L

1 ∂ρ ∂p ∂ρ ∂p δρ δp L − O ≤=×210−−11s 2 ; ρ222NM∂x ∂yy∂ ∂xQP ρ L

The sign ≤ indicates that these may be overestimated due to cancellation.

End of Chapter 7