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Vorticity and Pressure Equations and Their Applications

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Vorticity and Pressure Equations and Their Applications Thunderstorm Dynamics Houze sections 7.4, 8.3, 8.5, Refer back to equations in Section 2.3 when necessary. Bluestein Vol. II section 3.4.6. Review article "Dynamics of Tornadic Thunderstorms" by Klemp – handout. A. Equations of Motion Boussinesq approximated equations (neglecting friction and Coriolis force) du1'∂ p =− (1a) dtρ0 ∂ x dv1'∂ p =− (1b) dtρ0 ∂ y dw1'∂ p =− + B (1c) dtρ0 ∂ z where buoyancy B includes effects of air density and water loading. The prime is with respect to a horizontally homogeneous base state. Written in a vector form: 1 G dV 1 =− ∇p' +Bkˆ (3) dt ρ0 or G ∂V GG1 =−VV ⋅∇ − ∇ pBk' + ˆ . (4) ∂t ρ0 Verify it for yourself that GG GG⎛⎞VV⋅ G G VV⋅∇ =∇⎜⎟ − V ×() ∇× V (4a) ⎝⎠2 Therefore GGG G ∂⋅VVVp⎛⎞' G ˆ =−∇⎜⎟ + +VBk ×ω + (5) ∂t ⎝⎠2 ρ0 GGG ijk G G ∂∂∂ where ω =∇×V = is the 3Dvorticity vector . ∂∂∂xyz uvw 2 B. Vorticity Equation G ⎛⎞∂∂wvGG⎛⎞ ∂∂ uw ⎛⎞ ∂∂ vuG ω =−⎜⎟ijk +−⎜⎟ +− ⎜⎟. ⎝⎠∂∂yz⎝⎠ ∂∂ zx ⎝⎠ ∂∂ xy Derive 3-D vorticity equation by taking ∇ × (5) and recall ∇ ×∇( ) = 0 Æ G ∂ω G G =∇×()VBk ×ω +∇× ()ˆ (6) ∂t 1. Development of rotation (vertical vorticity) G To investigate the development of rotation in thunderstorms, look at the vertical component of vorticity ς = kˆ ⋅ω Æ ∂ς GGGG =⋅∇××kVˆˆˆˆ()ω +⋅∇× kBkkV () =⋅∇×× ()ω ∂t Note that kBkˆˆ⋅∇×()0 = (verify yourself) therefore buoyancy does not directly generate vertical rotation in thunderstorms! Buoyancy only generates horizontal vorticity which can be tilted into the vertical direction. Let ξ,ης and be the x, y and z component of vorticity, respectively. 3 GGG ijk G G GGG Vuvwvwiwujuvk×=ω =()()()ςη − + ξς − + ηξ − ξης and G G ∂∂ kVˆ ⋅∇×() ×ω =() wuξς − −() vw ς − η ∂∂xy ∂∂∂∂∂∂∂∂ςςςς⎛⎞uv w w G ∂ξ ∂∂ης =−uvw − − −ςξη⎜⎟ + + + (we have used ∇ ⋅≡ω + + =0 ) (7) ∂∂∂∂∂∂∂∂txyzxyxy⎝⎠ ∂∂∂xyz or ∂ς GGG =−VVw ⋅∇ςς − ∇ ⋅ + ω ⋅∇ (8) ∂t HHH The last term is called 'tilting' term. It turns horizontal vorticity into the vertical component through differential vertical motion. Making use of Boussinesq mass continuity equation GG∂w ∇⋅VV =∇ ⋅ + =0 H ∂z 4 Eq.(8) can be rewritten as ∂∂ς G w G =−Vw ⋅∇ςς + + ω ⋅∇ (9) ∂∂tzHH Local Advection Stretching Tilting change of term term term vertical Stretching term - if ς > 0 and w increases with height (stretching of air column), the term is positive Æ increases in ς . Vertical stretching corresponds to horizontal convergence – angular momentum conservation principle! Tilting term. If shear is in u only, ∂u > 0, the horizontal vorticity vector points in positive y direction. ∂z 5 ∂w ∂ς ∂∂∂wuw On the south side of updraft, > 0 Æ =η =>0 Æ the tilting creates positively vertical vorticity. ∂y ∂∂∂∂tyzy Similarly, one the north side, the tilting due to updraft motion creates negative vertical vorticity. 6 2. Generation of Horizontal Vorticity (rotation about horizontal axis) Where the environment has vertical shear, the environment contains horizontal vorticity – the vertical shear is a reservoir of horizontal vorticity. We saw earlier that through tilting of horizontal vortex tubes, horizontal vorticity can be transformed into vertical one – contributing to the thunderstorm rotation. We pointed earlier that buoyancy does not produce vertical vorticity, what about horizontal vorticity? Yes – it does. Remember horizontal vorticity generation at the gust front where there is strong horizontal buoyancy gradient? To investigate generation of vorticity about the horizontal, we need the equation for horizontal vorticity. Let's consider the y component of vorticity first: ∂∂uw η =−. ∂∂zx ∂∂ Take (.1)Eqa− (.1) Eqc, we obtain (show it yourself) ∂∂zx dBη ∂ =− , (10) dt∂ x therefore (apart from friction), horizontal gradient of buoyancy provides the only source of horizontal vorticity processed by an air parcel. 7 When the low-level air parcel trajectory has a significant parallel component to the gust front, the vorticity generation by the horizontal gradient can be significant, and the tilting of it into the vertical is believed to contribute significantly to the thunderstorm rotation too. 8 C. Pressure Perturbation Equation The rotational dynamics with supercell storms has a lot to do with the pressure perturbations created by the air flow. It is this effect that makes supercells special. ∂ ∂∂ Take ∇⋅(equation of motion), i.e., (1acd )++ (1 ) (1 ) : ∂∂∂xyz ∂∂⎡⎤uuu ∂ ∂ ∂ u1' ∂2 p 1sr term: ⎢⎥+++uvw =− 2 , ∂∂x ⎣⎦txyz ∂ ∂ ∂ρ0 ∂ x ∂∂⎡⎤vvv ∂ ∂ ∂ v1' ∂2 p 2nd term: ⎢⎥+++uvw =− 2 , ∂∂∂∂∂yt⎣⎦ x y zρ0 ∂ y ∂∂⎡⎤www ∂ ∂ ∂ w1' ∂2 pB ∂ 3rd term: ⎢⎥+++uvw =−+2 . ∂∂∂∂∂zt⎣⎦ x y zρ0 ∂∂ z z ∂uvw∂∂ Taking derivatives, combining terms and remembering + +=0, we have ∂∂∂xyz ⎡⎤222 1 2 ⎛⎞∂∂∂uvw⎛⎞ ⎛⎞ ∇=−p' ⎢⎥⎜⎟ +⎜⎟ + ⎜⎟ ρ0 ⎢⎥⎝⎠∂∂∂xyz⎝⎠ ⎝⎠ ⎣⎦ (11) ⎡⎤∂∂uv ∂∂ uw ∂∂ vw ∂ B −+++2⎢⎥ ⎣⎦∂∂yx ∂∂ zx ∂∂ zy ∂ z 9 Now let's partition winds between environmental wind and thunderstorm induced perturbation winds: uxyzt(, ,,)=+ uz () u '(,,,) xyzt vxyzt(, ,,)=+ vz () v '(,,,) xyzt wxyzt(, ,,)=+ 0 w '(, xyzt ,,) Substituting them into (11), we have ⎡⎤222 2 ⎛⎞∂∂∂uvwuvuwvw''⎛⎞ ⎛⎞ ' ∂∂∂∂∂∂ '''''' ∇=−p'222ρ0 ⎢⎥⎜⎟ +⎜⎟ + ⎜⎟ + + + ⎢⎥⎝⎠∂∂∂x ⎝⎠yz ⎝⎠ ∂∂∂∂∂∂ yxzxzy ⎣⎦ (12) ⎡⎤∂∂uw'' ∂∂ vw ∂ B −++ρρ00⎢⎥22 ⎣⎦∂∂zx ∂∂ zy ∂ z It's an elliptic diagnostic equation for pressure p'. Dividing the total perturbation pressure into pp''=+dyn p ' B =+pp''LNLB + p ' 2"nd term fluid extensionterm "+ shear term last term 10 Each part is attributed to certain terms on the right hand side of Eq.(12). Applications: 1. Updraft enhancement in rotating thunderstorms and cell splitting. We saw earlier that a strong updraft in an environment of significant vertical shear produces a pair of counter- rotating vortices. Consider p'NL term only. It can be verified that the 3 shear terms can be written in the following form: ∂∂uv'' ∂∂ uw '' ∂∂ vw '' 22++ 2 ∂∂yx ∂∂ zx ∂∂ zy 222 1''⎡⎛⎞⎛⎞∂∂vu ∂∂ vw ''⎛⎞ ∂∂ uw '' =+++++⎢ ⎜⎟ 2 ⎜⎟⎜⎟∂∂xy ∂∂ zy ∂∂ z x ⎣⎢⎝⎠⎝⎠⎝⎠ 22 2 ⎛⎞⎛⎞∂∂vu'' ∂∂ vw ''⎛⎞ ∂∂ uw ''⎤ −− −− −−⎜⎟⎥ ⎜⎟⎜⎟xy zy z x ⎝⎠⎝⎠∂∂ ∂∂⎝⎠ ∂∂⎦⎥ If we assume pure rotation (no div, deformation) and ignore extension terms (i.e., look at the effect of vertical rotation only), then 2 22ρρ00⎛⎞∂∂vu'' ∇=p''NL ⎜⎟ − =ς . 22⎝⎠∂∂xy 2 Since the left hand side is a Laplacian operator, ∇∝−p''NLp NL , therefore 11 2 p''NL ∝−ς . (13) Æ Both cyclonic and anticyclonic rotation produces negative pressure perturbation. The low pressure center is actually required to give a PGF that balances the centrifugal force! Negative p'NL is largest where rotation is the strongest, which is usually at the mid-levels of thunderstorms where the mesocyclone is found - the p'NL "low" there is up to 2-4 mb. Earlier figure shows that because of tilting, vertical rotation is strongest at the flanks of updraft, and negative p' at the mid-levels creates an upward pressure gradient force there that promotes new updrafts there – a dynamic cause for cell splitting. Now, let's consider the fluid extension terms, ⎡⎤222 2 ⎛⎞∂∂∂uvw''⎛⎞ ⎛⎞ ' ∇=−p'...0ρ0 ⎢⎥⎜⎟ +⎜⎟ + ⎜⎟ +< ⎣⎦⎢⎥⎝⎠∂∂∂xyz⎝⎠ ⎝⎠ 222 ⎡⎤⎛⎞∂∂∂uvw''⎛⎞ ⎛⎞ ' p'∝+++>ρ0 ⎢⎥⎜⎟ ⎜⎟ ... 0 xyz⎜⎟ ⎣⎦⎢⎥⎝⎠∂∂∂⎝⎠ ⎝⎠ So positive p' is largest where stretching is largest. This occurs at the low and high levels. 12 So we have p'NL > 0 H p'NL < 0 L ⇑ vertical PGF p'NL > 0 H Therefore the nonlinear pressure perturbation due to shear and stretching creates additional upward lifting (pressure gradient) force at the lower atmosphere that enhances the updraft beyond that based on buoyancy! Therefore, supercell storms tend to be stronger than regular storms, given the same amount of CAPE. Rule of thumb: 1mb VPGF over 1 km ~ same forcing as 3°C of buoyancy. 13 14 15 2. Preferred enhancement of right-moving or left-moving storm Consider the linear p'dyn term: G 2 ⎡⎤∂∂uw'' ∂∂ vw ∂ V ∇=−p'2ρρ00⎢⎥ + =− 2 ⋅∇w ' ⎣⎦∂∂zx ∂∂ zy ∂ z G ∂V so p''∝⋅∇w . (14) ∂z For unidirectional shear x ∂∂uw' p'0∝> on the west/upshear flank of updraft ∂∂zx 16 ∂∂uw' p'0∝< on the east/downshear flank of updraft and p' largest ∂∂zx at the mid-levels p' > 0 p'<0 Æ new cell growth on the downshear flank If the hodograph is clockwide curved, v u ∂∂vw' then we also have to consider term in (14). ∂∂zy 17 L y ∂∂vw' At low levels, , produces x ∂∂zy H ∂∂uw' At the mid-levels, produces H L ∂∂zx H ∂∂vw' At the upper levels produces ∂∂zy L Æ there is a upward vertical PGF on the right flank of the storm (downward PGF on the left flank) Æ new cell growth is enhanced to the right, rotating updraft becomes a 'right mover'. 18 19 20 21 Summary: The nonlinear-shear effect that promotes new or continued cell growth on the flanks of (alongside) the old cell; The linear effect of tilting biases the cell movement toward the right (left) if the environmental hodograph is curved in a clockwise (counterclockwise) manner; Unidirectional shear promotes storms that split, with each member of the split pair having components of motion normal to the shear vector and opposite to each other. New buoyant updrafts form off the axis of the shear because upward-directed perturbation pressure gradients induce upward accelerations there and lift air to its LFC. Owing to the low-level convergence associated with upward-moving air, vorticity increases through the stretching of the existing vorticity and is advected upward by the updraft: Right movers tend to develop cyclonic rotation, while left movers tend to develop anti-cyclonic rotation. The cyclonic vorticity produced in storms that grow in an environment of clockwise turning shear is not due to the Earth's rotation. 22.
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