Potential Vorticity (PV) Dynamics and
Models of Zonal Flow Formation
Pei-­‐Chun Hsu, Patrick Diamond CASS, UCSD APS-­‐DPP, Nov 16-­‐20, 2015 1 Outline I.  Introduc<on II.  Non-­‐perturba<ve MFT of turbulent relaxa<on and PV transport –  structure of PV flux (1)(2) i. Selec<ve decay model: minimum enstrophy principle ii. PV-­‐avalanche model: symmetry principles II.  Perturba<on theory of PV flux –  transport coefficients (1) i. Modula<onal instability ii. Parametric instability III.  Zonal flow forma<on in the presence of ambient mean shear (3) IV.  Summary (1) Pei-­‐Chun Hsu and P. H. Diamond, Phys. Plasmas, 22, 032314 (2015) (2) Pei-­‐Chun Hsu, P. H. Diamond, and S. M. Tobias, Phys. Rev. E, 91, 053024 (2015) (3) Pei-­‐Chun Hsu and P. H. Diamond, Phys. Plasmas, 22, 022306, (2015) 2 Introduction:
unifying concept
PV mixing & ZF forma5on •  PV conserva<on dq
= 0 dt
GFD: Plasma: Quasi-­‐geostrophic system Hasegawa-­‐Wakatani system q = ∇ 2ψ + β y
rela<ve vor<city q = n − ∇ 2φ
planetary vor<city (
2
)
Physics: Δy
→
Δ ∇
ψ
à ZF! ion vor<city density (guiding center) (polariza<on) (
)
Physics: Δr
→
Δn
→
Δ
∇
2 φ à ZF! •  Zonal Flow forma<on. ∂
∂y
Taylor’s iden<ty υ ! y q! = − υ ! x υ ! y à PV flux fundamental to ZF forma5on 3 Introduction:
physical systems
turbulence quasi-­‐geostrophic dri?-­‐wave force Coriolis Lorentz velocity geostrophic magne<c energy linear waves Rossby waves dria waves conserved PV q = ∇ 2ψ + β y
inhomogeneity characteris<c scale fast frequency turbulence β
LD ≈ 10 6 m
f ≈ 10 −2 s −1
q = n − ∇ 2φ
∇n, ∇T
ρ s ≈ 10 −3 m
ω ci ≈ 108 s −1
usually strongly driven not far from marginal Reynolds number Re >>1 Re ~ 10-­‐102 zonal flows Jets, zonal bands sheared E x B flows role of zonal flows transport barriers L-­‐H transi<on 4 PV flux I
General Structure
Non-­‐perturba5ve approaches -  PV mixing in space is essen<al in ZF genera<on. 2
Taylor iden<ty: υ y ∇ φ = −∂ y υ yυ x
vor<city flux Reynolds force Key: How represent inhomogeneous PV mixing most treatment of ZF: -­‐-­‐ perturba<on theory -­‐-­‐ modula<onal instability (test shear + gas of waves) ~ linear theory -­‐> physics of evolved PV mixing? -­‐> something more general? General structure of PV flux? àrelaxa<on principles! non-perturb model 1: use selec<ve decay principle What form must the PV flux have so as to dissipate enstrophy while conserving energy? non-perturb model 2: use joint reflec<on symmetry What form must the PV flux have so as to saGsfy the joint reflecGon symmetry principle for PV transport/mixing? 5 non-perturb model 1
General principle: selec5ve decay •  2D turbulence conserva<on of energy and poten<al enstrophy à  dual cascade à  Minimum enstrophy state inverse energy cascading forward enstrophy cascading •  eddy turnover rate and Rossby wave frequency mismatch are comparable ∂ω !
+ u ⋅ ∇ω + βυ = 0
∂t
U
U2
βU
2
LT
L
U
à Rhines scale LR ~ β
Rhines scale wave wave zonal flow forcing 6 non-perturb model 1
Using selec5ve decay for flux minimum enstrophy Taylor relaxa5on analogy relaxa5on (J.B. Taylor, 1974) (Bretherton & Haidvogel 1976) turbulence dual cascade 3D MHD conserved quan<ty (constraint) total kine<c energy dissipated quan<ty (minimized) fluctua<on poten<al enstrophy magne<c energy minimum enstrophy state Taylor state flow structure emergent force free B field configura<on final state structural approach •
2D hydro ∂
Ω < 0 ⇒ ΓE ⇒ Γq
∂t
global magne<c helicity ∂
EM < 0 ⇒ Γ H
∂t
flux? what can be said about dynamics? à  structural approach (Boozer): What form must the helicity flux have this work): What form must the PV flux have so sao s ats o to dissipate m
agneGc-­‐energy hile conserving helicity? enstrophy while cw
onserving energy? General principle based on general physical ideas à useful for dynamical model 7 non-perturb model 1
PV flux à PV conserva<on mean field PV: selec<ve decay ∂ q
+ ∂ y υ y q = ν 0∂2y q
∂t
Γ q : mean field PV flux Key Point: form of PV flux Γq which dissipates enstrophy & conserves energy 2
∂ φ )
(
à energy conserved E = ∫ 2
y
∂E
=
∂t
∫
φ ∂yΓ q = − ∫ ∂y φ Γ q
à enstrophy minimized Ω = ∫
⇒ Γq =
q
2
∂yΓ E
∂y φ
2
&∂ q )
∂Ω
++ Γ E
= − ∫ q ∂ y Γ q = − ∫ ∂ y (( y
∂t
∂
φ
' y
*
&∂ q )
∂Ω
++
< 0 ⇒ Γ E = µ∂ y (( y
∂t
' ∂y φ *
parameter TBD υx
& ∂ q )0
1
++2
⇒ Γq =
∂ y /µ∂ y (( y
∂y φ
/.
' ∂ y φ *21
general form of PV flux 8 non-perturb model 1
Structure of PV flux )
#∂ q
1
Γq =
∂ y +µ ∂ y %% y
υ x +*
$ υx
) # q ∂ q ∂2 q
&,
1
y
((. =
∂ y +µ %%
+ y
2
υ x +* $ υ x
υx
'.-
diffusion parameter calculated by perturba<on theory, numerics… &,
(.
(.
'-
diffusion and hyper diffusion of PV <-­‐-­‐> usual story : Fick’s diffusion relaxed state: ∂y q
Homogeniza<on of à allows staircase υx
characteris5c scale ℓ c ≡
υx
∂y q
ℓ > ℓ c : zonal flow growth ℓ < ℓ c : zonal flow damping (hyper viscosity-­‐dominated) Rhines scale LR ~
U
β
ℓ > LR : wave-­‐dominated ℓ < LR : eddy-­‐dominated 9 non-perturb model 1
PV staircase ∂ y q PV g radient large relaxed state: homogeniza<on of à where zonal flow large υ
x
à  Zonal flows track the PV gradient à PV staircase •  Highly structured profile of the staircase is reconciled with the homogeniza<on or mixing process required to produce it. •  Staircase may arise naturally as a consequence of minimum enstrophy relaxa<on. 10 non-perturb model 1
What sets the “minimum enstrophy” •  Decay drives relaxa<on. The relaxa<on rate can be derived by linear perturba<on theory about the minimum enstrophy state q = qm (y) + δ q(y, t)
φ = φ m (y) + δφ (y, t)
∂ y qm = λ∂ yφ m
δ q(y, t) = δ q0 exp(−γ rel t − iω t + iky)
" k 4 + 4 λ k 2 + 3λ 2 8q 2 (k 2 + λ ) %
'
γ rel = µ $$
− m
2
4
'
υx
υx
#
&
" 4q k 3 +10q k λ 8q 3 k %
m
ω k = µ $$ − m
− m 5 ''
3
υx
υx &
#
>0 relaxa<on •  The condi<on of relaxa<on (modes are damped): 8qm2
8qm2
2
− 3λ ⇒
> 3λ à Relates q m2 with ZF and scale factor γ rel > 0 ⇒ k >
2
2
υx
υx
⇒ υx
2
<
3λ
8qm2
ZF can’t grow arbitrarily large 2
⇒ 8 qm2 > υ x 3λ the ‘minimum enstrophy’ of relaxa<on, related to scale 11 non-perturb model 1
Role of turbulence spreading •  Turbulence spreading: tendency of turbulence to self-­‐scater and entrain stable regime •  Turbulence spreading is closely related to PV mixing because the transport/mixing of turbulence intensity has influence on Reynolds stresses and so on flow dynamics. •  PV mixing is related to turbulence spreading ∂E
=
∂t
∫
φ ∂yΓ q = − ∫ ∂y φ Γ q
⇒ Γq =
∂yΓ E
∂y φ
•  The effec<ve spreading flux of turbulence kine<c energy Γ E = − ∫ Γ q υ x dy = − ∫
+
%∂ q
1
∂ y -µ∂ y '' y
υ x -, & υ x
(.
%∂ q
**0 υ x dy = µ∂ y '' y
)0/
& υx
(
**
)
àthe gradient of the ∂y⟨q⟩/⟨vx⟩, drives spreading à the spreading flux vanishes when ∂y⟨q⟩/⟨vx⟩ is homogenized 12 non-perturb model 1
Discussion •  PV mixing forward enstrophy cascade hyper-­‐viscosity à How to reconcile effec<ve nega<ve viscosity with the picture of diffusive mixing of PV in real space? •  A possible explana<on of up-­‐gradient transport of PV due to turbulence spreading PV mixing Turbulence spreading Weaker turbulence intensity (enstrophy) Stronger turbulence intensity (enstrophy) 13 non-perturb model 2
PV-­‐avalanche model – beyond diffusion : PV profile : self-­‐organized state
δ q > 0 δ q = q − q : d evia<on
0
mixing Γ[δ q]
Γ[δ q]
Γ[δ q] : PV flux δq < 0
More general form of PV flux pulse propaga<on y •  avalanching: tendency of excita<on to propagate in space via local gradient change •  Joint-­‐reflec<on symmetry: Γ [δq] invariant under y à -­‐y and δq à -­‐δq Key Point: n
2l
m
3
form o
f P
V fl
ux w
hich s
a<sfies à Γ [δ q ] = ∑α l (δ q ) + ∑ β m (∂ yδ q ) + ∑γ n ∂ yδ q +...
l
m
n
joint-­‐reflec<on symmetry •  large-­‐scale proper<es : higher-­‐order deriva<ves neglected small devia<ons : higher-­‐order terms in δq neglected α
2
3
à Simplest approxima<on: Γ [δ q ] = (δ q ) + β∂ yδ q + γ∂ yδ q
2
(
)
general form of PV flux 14 non-perturb model 2
Γ [δ q ] =
•  PV equa<on: α
2
(δ q) + β∂yδ q + γ∂3yδ q
2
2
y
4
y
∂tδ q + αδ q∂ yδ q + β∂ δ q + γ∂ δ q = 0
Kuramoto-­‐Sivashinsky type equa<on diffusion and hyper diffusion of δq Non-­‐linear convec<on of δq •  Avalanche-­‐like transport is triggered by devia<on of PV gradient à PV devia<on implicitly related to the local PV gradient δ q → ∂ y q
à transport coefficients (func<ons of δq) related to the gradient D(δ q) → D(∂ y q)
à gradient-­‐dependent effec<ve diffusion Γ q ~ −D(∂ y q)∂ y q → −D(δ q)δ q
à Convec<ve component of the PV flux can be related to a gradient-­‐
dependent effec<ve diffusion Γ [δ q ] ~ δ q 2 → −D(δ q)δ q
D(δ q) → D0δ q
15 PV flux II
Setting the Coefficients
Perturba5on theory General structure of PV flux à relaxa<on principles Transport coefficients of PV flux à perturba<on theory (only analy<cal solu<on) •  The evolu<on of perturba<on (seed ZF) as a way to look at PV transport Seed ZF Modula<on (nonlocal) Fluctua<ons (broad spectrum) υ! xυ! y
Inverse cascade/ PV mixing (local) Modula<onal instability (large scale pumping) requires small scale PV mixing 16 perturbation theory 1
– broadband
Revisi5ng modula5onal instability ZF evolu<on determined by Reynolds force kk
∂
∂
∂
δVx = −
υ xυ y = ∑ x 4 y N k
∂t
∂y
∂y k< k
vor<city flux 2
N k = k 2 ψ k ω k is wave ac<on density, for Rossby wave and dria wave, it is propor<onal to the enstrophy density. Nk is determined by WKE: ∂N
∂(kxδVx ) ∂N 0
+ υ g ⋅ ∇N + δω k N =
∂t
∂y
∂ky
➜  Turbulent vor<city flux derived "k k
∂
2
δVq = ∂ yδVq ∑$$
∂t
k # k
2
x y
4
%
δω k
∂N 0
''
2
2
& (ω q − q ⋅ υ g ) + δω k ∂k y
κ(q) q: ZF wavenumber ∂
δVq = ∂ 2yκ (q) δVq
∂t
κ(q) ≠ const ➜  scale dependence of PV flux ➜  non-­‐Fickian turbulent PV flux 17 perturbation theory 1
•  A simple model from which to view κ (q): -  Defining MFP of wave packets as the −1
−1
cri<cal scale qc ≡ υ gδω k
-  keeping next order term in expansion of response δω k
1 $ q2 '
q >> q ⇒
≈
&1− )
(qυ g )2 + δω k2 δω k % qc2 (
−1
−1
c
➜  zonal growth evolu<on: ∂tδVx = D∂2yδ Vx − H∂4yδVx
➜  nega<ve viscosity and posi<ve hyper-­‐viscosity D=
k x2 k y∂N 0
<0
δω k k 4 ∂ky
∑
k
k x2 k y∂N 0
H = −∑ q
>0
4
δω k k ∂ky
k
< 0 −2
c
Transport coefficients (viscosity and hyper-­‐viscosity) for relaxa<on models: * $
'2
∂ υx
q
∂
q
∂
q
1
y
)/
= Γq =
∂ y ,µ & −
+ y
2
, & (∂ φ ) ∂ y φ )/
∂t
∂y φ
y
(.
+ %
18 perturbation theory 1
Discussion of D and H •  Roles of nega<ve viscosity and posi<ve hyper-­‐viscosity (Real space) ∂
δVx = D∂2yδVx − H∂4yδVx
∂t
2 2
2 2
∂ 1 2 2
2
δ
V
d
x
=
−D
∂
δ
V
d
x
−
H
∂
δ
V
∫ x
∫ ( y x)
∫ ( y x) d x
∂t 2
D < 0 ⇒ γ q,D > 0 ZF growth (Pumper D) Energy transferred to large scale ZF H > 0 ⇒ γ q,H < 0 ZF suppression (Damper H) ➜  D, H as model of spa<al PV flux beyond over-­‐simplified nega<ve viscosity 2 sets the cut-­‐off scale D = Hq
⇒ lc2 =
H
D
Minimum enstrophy model )
# ∂ q &,
1
((. ⇒ ℓ ≡
Γq =
∂ y +µ ∂ y %% y
c
∂y φ
∂
φ
+*
$ y
'.-
υx
∂y q
 >  c : ZF energy growth à D process dominates at large scale  <  c : ZF energy damping à H process dominates at small scale 19 perturbation theory 2
– narrowband
Parametric instability Pseudo-­‐fluid (wave packets) model mean free path of wave packets Pei-­‐Chun Hsu and P. H. Diamond, Phys. Plasmas, 22, 032314 (2015) 20 perturbation theory 2
υ! xυ! y = ∫ υ gy kx N k d 2 k ≅ Vyw Pxw pseudo-­‐momentum flux •  ZF evolu<on: ∂
υ = − ∂ V
w P w x
y
x
∂t
∂y
•  pseudo-­‐fluid evolu<on: ∫ (WKE )υ
n
gy
d 2k
w
2 β kx2 " 4ky
$1− 2
k4 #
k
2
a=
∂
w ∂ V w = −a
υ $
⇒ V y w +V
y
y
x
∂t
∂y
∫
%
'N k dk
&
∫N
k
dk
inviscid Burgers’eq. source: zonal shear à  ZF growth rate in monochroma<c limit: " 4k 2y %
2
2 2
γ q = q k x ϕ k $$1− 2 ''
k &
#
Convec<ve transport coefficients for PV-­‐avalanche model: -­‐-­‐ Instability ( γ
q real ) à k x2 > 3k y2
-­‐-­‐ γ q ∝
q
à convec<ve instability 21 •  Minimum enstrophy * $
'2
∂ υx
q
∂
q
∂
q
1
y
y
)/
= Γq =
∂ y ,µ & −
+
2
, & (∂ φ ) ∂ y φ )/
∂t
∂y φ
y
(.
+ %
α
2
δ
q
( ) + β∂yδ q + γ∂3yδ q
2
•  PV-­‐avalanche Γ [δ q ] =
•  Modula<onal instab. ∂tδVx = −q 2 DδVx + q 4 H δVx
•  Parametric instab. 2 2
x
γ q = q k ϕk
2
" 4k 2y %
$$1− 2 ''
k &
#
22 III) Mul5-­‐scale shearing effects •
Mo<va<on: coexistence of mean shear and zonal flow shear (different roles) à L-­‐H transi<on, the solar tachocline Generic problem: interac<on between different scale shearing fields mean shear
•
zonal flow
wave packet
y
Vx
•
Important issue: how mean shear affects the PV flux and ZF genera<on Turbulence Free Energy Source (gradients) Shearing Shearing Modula<onal Instability Zonal Flows Decorrela<on (this work) Mean Shears Pei-­‐Chun Hsu and P. H. Diamond, Phys. Plasmas, 22, 022306, (2015) 23 Modula5onal instab. w/ mean shear Momentum flux -­‐> Reynolds stress -­‐> wave ac<on < υ xυ y >= −∑
k
kx ky
Nk
k4
•  mean shear in WKE: ∂t N k + υ gy∂ y N k − kx Vx # ∂ky N k + δω k N k = kxδVx '∂ky N 0
(shearing rate Ω) decrease υ gy ∝ Ω−3
•  Ray trajectory refrac<on: dky
∂
= − (ω + kxVx ) ;Vx = Vx + V!x
dt
∂y
2 β kx ky
dy
= υ gy =
dt
k4
mean shear Non-­‐linear −∂ y Vx = Ω
diffusion ky (t) = ky (0) + kxΩt
Seed ZF Kx ky wave trajectory is distorted by surrounding mean shears smaller scale &
β#1
1
(
y(t) = y(0) + e(t), e(t) = %% 2 − 2
2(
Ω $ k0 kx + (k 0 y + kx Ωt) '
excursion inhibited 24 Modula5onal instab. w/ mean shear •  characteris<c method (shearing frame) ky = ky (0) + kxΩt


original frame shearing frame The
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∂t N k + υ gy∂ y N k − kx Vx # ∂ky N k + δω k N k = kxδVx '∂ky N 0
y = y(0) + e(t)
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-­‐
-­‐
Solving Green’s func<on in shearing frame Changing variables back to original frame The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart
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•  strong mean shear limit ( ): The image cannot be
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1/3
$ 3 '
2 !
γ q , υ! y ∇ ψ ∝ &
2)
% δω k Ω (
→ Mean shear reduces ZF growth, PV flux ~Ω-­‐2/3 → scaling of PV flux in strong mean shear 25 Summary •  Inhomogeneous PV mixing is iden<fied as the fundamental mechanism for ZF forma<on. This study offered new approaches to calculaGng spaGal flux of PV. •  The general structure of PV flux is studied by two non-­‐perturbaGve relaxa<on models. In selec<ve decay model, PV flux contains diffusive and hyper-­‐
diffusive terms. In PV-­‐avalanche model, PV flux contains another convec<ve term, which can be generalized to an effec<ve diffusive transport. •  The transport coefficients are derived using perturba<on theory. In modula<onal instability analysis for a broadband spectrum, a nega<ve viscosity and a posi<ve hyper-­‐viscosity, which represents ZF satura<on mechanism, are derived. In parametric instability analysis for a narrow spectrum, a convec<ve transport coefficient is obtained. •  Important issues addressed in our models includes PV staircase, turbulence spreading, avalanche-­‐like transport, characterisGc scales. •  The effect of the mean shear on ZF forma<on is studied. ZF growth rate and the PV flux are shown to decreases with mean shearing rate. Framework of PV transport for systems with mulG-­‐scale shearing fields is established. 26 Acknowledgement This material is based upon work supported by the U.S. Department of Energy Office of Science under Award Number DE-­‐FG02-­‐04ER54738. 27