L.M. Smith. with collaborators. M. Remmel, PhD student, UW-Madsion soon to be VIGRE Assistant Professor at UC-Davis

Transcription

1 p. 1/4 New PDEs for Rotating Stratified Fluid Flow L.M. Smith University of Wisconsin, Madison with collaborators M. Remmel, PhD student, UW-Madsion soon to be VIGRE Assistant Professor at UC-Davis J. Sukhatme, Assistant Professor Indian Institute for Science, Bangalore

2 p. 2/4 Pictures of Atmospheric Waves Roll clouds in the jet stream over Saudi Arabia/Red Sea. Zoom out of the jet stream moving from left to right. From

3 p. 3/4 A complicated interplay of waves and coherent structures Left: Gulf Stream leaving the east coast of North America Right: Hurricane Ivan From

4 p. 4/4 Questions: What is the role of wave interactions for the generation of coherent, large-scale JETS, VORTICES and LAYERS? (dry dynamics first) Can we construct a framework for a complete understanding of the role of inertia-gravity waves in geophysical flows?

5 p. 5/4 The 3D rotating Boussinesq equations on an f-plane Conservation laws for vertically stratified flow rotating about the vertical ẑ-axis: momentum : Du Dt + fẑ u = P Nθẑ + ν 2 u mass : u = 0 energy : Dθ Dt Nw = κ 2 θ, θ = g ρ Nρ o D Dt = t + u f = 2Ω, Ro = U fl ρ = ρ o bz + ρ, ρ ρ o, bz, N 2 = gb ρ o, Fr = U NH

6 p. 6/4 Rossby and Froude numbers (Pedlosky, 1986) Ro 0.14 for typical synoptic-scale winds at mid-latitudes. e.g Ro = 0.14 uses U 10 m s 1, L 1000 km f 10 4 s 1 Ro 0.07 in the western Atlantic e.g. Ro = 0.07 uses U 5 cm s 1, L 100 km Typical values of N/f are N/f 100 in the stratosphere and N/f 10 in the oceans.

7 p. 7/4 Solutions in the unforced, linear, inviscid limit The solution form [ ( [u(x,t;k);θ(x,t;k)] = φ(k) exp i k x σ(k) t )] R + c.c. with eigenmodes φ(k) and eigenvalues σ(k). Wave modes φ + (k) and φ (k) with σ ± (k) = ± (N2 k 2 h + f2 k 2 z) 1/2 k A non-wave (vortical) mode φ 0 (k) with σ 0 (k) = 0.

8 p. 8/4 Eigenmode representation for nonlinear flows Since φ s (k), s = ±, 0 form an orthogonal basis u(x,t) = k and the equations become [ ( b s (t;k)φ s (k)exp i k x σ s (k) t )] R s t b s k = C skspsq kpq s p,s q [ ( ) ] t b s p b s q exp i σ sk + σ sp + σ sq R Exact and near resonances dominate for R 1: σ sk + σ sp + σ sq R

9 p. 9/4 Reduced Models Reduced models resulting from restriction of the sum t b s k = C skspsq kpq s p,s q [ ( ) ] t b s p b s q exp i σ sk + σ sp + σ sq R automatically conserve energy because each triad (k, p, q) satisfies: C s ks p s q kpq + C s ps q s k pqk + C s qs k s p qkp = 0

10 p. 10/4 Success with models of near resonances: σ sk + σ sp + σ sq < R Full Near resonances (12%) Smith & Lee 2005, but not a PDE!

11 p. 11/4 Symbolically The full equations: 0 [00] [0+] [0 ] [++] [+ ] [ ] + [00] [0+] [0 ] [++] [+ ] [ ] [00] [0+] [0 ] [++] [+ ] [ ] where 0, +, - represent vortical and wave linear eigenmodes.

12 p. 12/4 Reduced Models: restrictions of the sum; always energy conserving! QG (vortical mode interactions only), Salmon (1981): 0 [00] GGG (wave modes only): + [++] [+ ] [ ] [++] [+ ] [ ]

13 p. 13/4 and everything in between, e.g. PPG (add to QG interactions involving exactly 1 wave): 0 [00] [0+] [0 ] + [00] [00] Muraki, Snyder, Rotunno (1999), McIntyre & Norton (2000)

14 p. 14/4 QG and PPG Rotating Shallow Water (RSW) Equations QG: q/ t + J(ψ,q) = 0 PPG : 2 χ t 2 R = 2J( A x, A ), (1) y q t + J(ψ,q) + χ q + u q x + v q y + q 2 χ = 0, (2) 2 R t c 2 4 χ + f 2 2 χ = fj(a,q) (3) q = ( 2 f2 gh )ψ, u = χ x ψ y, v = χ y + ψ y 2 R = 2 (fψ gh) geostrophic imbalance A (f 2 c 2 2 ) 1 c 2 Q, c 2 = (gh) 1/2

15 p. 15/4 Succes in RSW decay, Ro=0.4, Fr = 0.25, divergence-free unbalanced i. QG x PPG x RSW x

16 p. 16/4 Centroid in RSW decay; divergence-free unbalanced i.c. Centroid (k) QG PPG P2G QVD Centroid (k) Linear QG PPG P2G QVD Time Time Ro = 0.4, Fr = 0.25 Ro = 0.25, Fr = 0.2 Cent(k) = ( k k( u k 2 + v k 2 )/ k ( u k 2 + v k 2 )

17 p. 17/4 Vorticity skewness in RSW decay; divergence-free unbalanced i.c PPG P2G QVD Skewness QG PPG P2G QVD Time Ro = 0.4, Fr = 0.25

18 p. 18/4 Compare to McIntyre & Norton PV inversion (MN1, MN2) 0 RSW PPG MN1 MN skewness time

19 p. 19/4 Why does this method work? Some near resonances included Exact dispersion relation is preserved Perturbative approaches to correct QG usually do neither!

20 p. 20/4 back to the Boussinesq equations Now show how to derive the PDE hierarchy for Boussinesq, in physical space, without using linear eigenmode expansion

21 p. 21/4 In physical space, change variables Introduce a velocity potential and streamfunction: u = χ x ψ y + u(z), v = χ y + ψ x + v(z), = horizontal avg and physical variables: q = 2 h ψ f N θ z linear potential vorticity R = N f θ + ψ z geostrophic imbalance and an operator: O = ( 2 h + f2 N 2 zz) with Q = O 1 q, R = O 1 R

22 p. 22/4 ===> an equivalent form of rotating Boussinesq: q t + ẑ (u u) f N z[(u )θ] = 0 f 2 h R t N 2 Ow + N 2 h [(u )θ] + f z(ẑ (u u)) = 0 2 w t + f 2 h R + 2 h (u w) z( h (u u h )) = 0 u t fv + z(uw) = 0, v t + fu + z(vw) = 0. θ = f N ( 2 h R zq), ψ = Q + f2 N 2 z R, χ = 2 h zw

23 p. 23/4 Symbolically The full equations: q [qq] [qr] [qw] [RR] [Rw] [ww] R [qq] [qr] [qw] [RR] [Rw] [ww] w [qq] [qr] [qw] [RR] [Rw] [ww] where q is linear PV; R, w contain wave information.

24 p. 24/4 From here, we can get to... Reduced PDEs (viscous terms not included), e.g., QG results from q [qq] ( ) t + u h q = 0, q = 2h f2 ( 2 ) + N 2 z 2 ψ(x, t) 2 h = 2 x y 2, u h = ẑ ψ, θ = f N ψ z

25 p. 25/4 or the antithesis of QG energy conserving GGG: R [RR] [Rw] [ww] w [RR] [Rw] [ww]

26 p. 26/4 or the antithesis of QG GGG in physical space: f 2 h R t 2 w t q t = 0 N 2 Ow + N 2 h [(u )θ ] + f z (ẑ (u u ) = 0 + f 2 h R + 2 h (u w) z ( h (u u h)) = 0 u t fv + z(u w) = 0 v t + fu + z(v w) = 0

27 p. 27/4 with definitions: u χ x f2 N 2 R zy + u(z), v χ y + f2 N 2 R zx + v(z) w w, θ f N 2 h R to eliminate interactions involving q from the nonlinear terms.

28 p. 28/4 and everything in between, e.g. PPG (add to QG interactions involving exactly 1 wave): q [qq] [qr] [qw] R [qq] w [qq]

29 p. 29/4 What can we learn from GGG? Consider 2 parameter regimes: Strongly/purely stratified flow with Fr 1 3-wave exact resonances to explain spectral scaling of ocean spectra McComas & Bretherton (1977), Lvov, Polzin, Tabak (2004) Bu = (NH/fL) 2 = O(1), N/f > 1, H/L < 1 Babin, Mahalov, Nicolaenko (2002) Embid & Majda (1996, 1998) derive QG dynamics in the limit Ro Fr = ǫ 0.

30 p. 30/4 Oceanic internal gravity wave spectra 3D strongly stratified flow with N/f 10 Garrett-Munk (observations) E(k h,k z ) = 2fNE (k z /k z) π[1 + (k z /k z)] 5/2 (f 2 k 2 z + N 2 k 2 h ) E and k z are fitting parameters Hydrostatic, high-wavenumber, high-frequency form E(k h,k z ) k 2 h k 3/2 z

31 p. 31/4 Previous theory Weak turbulence theories keeping only 3-wave exact resonances McComas & Bretherton, 1977 Lvov, Polzin & Tabak 2004, 2006: the kinetic equations for hydrostatic, purely stratified flow admit a family of power-law solutions E(k h,k z ) k α h kβ z What about near and non-resonant 3-wave interactions?

32 p. 32/4 Purely stratified, non-hydrostatic GGG = waves only 2 w t u(z) t N 2 h θ t 2 h ψ t = 0 N 2 2 h w + N 2 h (u θ ) = 0 + N 2 h θ + 2 h (u w) z h (u u h ) = 0 + z (u w) = 0, v(z) t + z (v w) = 0, θ(z) t = 0 u = χ x + u(z), v = χ y + v(z), w = w, θ = θ θ(z)

33 p. 33/4 What can we learn about response to unbalanced, high-frequency forcing? Waite & Bartello (2006) Growth of vertically sheared, horizontal flows: VSHF (VSHF, V, W) vs. (VSHF, W, W) VSHF cannot be generated by exact resonances We never force VSHF directly

34 VSHF p. 34/4

35 p. 35/4 Back to physical space: Generation of VSHF in GGG: (VSHF,±, ±): t u = z (χ x w), t v = z (χ y w) In the full model we also have: (VSHF, 0, ±) t u = z (ψ x w), t v = z (ψ y w)

36 p. 36/4 Energies in time 9 GGG, Fr = unbalanced force full model, Fr = energies 5 4 energies time time blue 128 3, red 162 3, black 192 3, green 256 3

37 p. 37/4 Compare growth of VSHF 8 GGG(RED)vsUF(BLUE) Fr.05 energies time red GGG; blue unbalanced forcing, full model clearly t u = z χ x w, t v = z χ y w most important

38 p. 38/4 GGG Spectra (with caution), Fr = 0.05, E(k) Wavespec GGG Fr= (yellow is same as & covered by thick black) Straight Lines ~k E ± z (k z ) Straight Line k 2 GGG Fr= E z (±) (kz T= k k z

39 p. 39/4 Easy to generalize! hydrostatic GGG rotating GGG earlier equations weak turbulence theory needs a homogeneous dispersion relation!

40 p. 40/4 Conclusions/Discussion New PDE hierarchy from QG or GGG to RBE and everything in between. Can be derived (i) by eigenfunction expansion, or (ii) directly in physical space Advantages: (i) preserve dispersion relation; (ii) include some near resonances Scaling of ocean spectra is a physical problem for which GGG may improve upon previous theory. Hope is for other mathematicians to be interested!