Magneto-geostrophic adjustment in rotating shallow water ......2015/09/21 · This is small Rossby-number limit of the lower branch of the dispersion relation of magneto-inertia-gravity

MotivationsRotating Shallow Water Magnetohydrodynamics (RSWMHD) and its 1.5D version

Quasigeostrophy and quasimagnetogeostrophy in RSWMHDNumerical simulations of the adjustment

Conclusions and Discussion

Magneto-geostrophic adjustment in rotatingshallow water magnetohydrodynamics

V. Zeitlin

1Laboratory of Dynamical Meteorology, University P. and M. Curie, Paris, France

Leverhulme Workshop, Oxford, September 2015

V. Zeitlin Magneto-geostrophic adjustment




GFD vs MHD

Main questionDoes fundamental in GFD notion of geostrophic adjustmentapply to MHD?

Main differenceAbsence of the spectral gap (f -plane) in MHD, because Alfvènwaves can have arbitrarily small frequencies.





Observations

MHD analog of shallow water

Fundamentals of geostrophic adjustment were first establishedin Rotating Shallow Model (RSW). MHD analog: rotatingshallow water magnetohydrodynamics (RSWMHD)

Useful simplificationAll most important features of geostrophic adjustment can beunderstood within a "1.5D" RSW. Try "1.5D" RSWMHD?





Geostrophy vs magneto-geostrophy

Standard RSW at small Rossby numbers: close to thegeostrophic balance⇒ QG approximation for slow balancedmotions. Universality of geostrophic adjustment - justification ofthe QG.RSWMHD at small Rossby and magnetic Rossby numbers:close to the magneto-geostrophic balance. Main newdynamical entity in RSWMHD: Alfvèn waves.

Questions:How the presence of Alfvèn waves modify the process ofgeostrophic adjustment (magneto-geostrophicadjustment)?What is the balanced dynamics in RSWMHD and is theQG approximation, heuristically used in MHD, valid?





RSWMHD1.5D RSWMHD and its properties

Rotating Shallow Water Magnetohydrodynamics

Derivation: vertical averaging of MHD equations.Hypotheses: magneto-hydrostatic equilibrium andmean-field approximation.Describes: motion of a thin layer of a magneto-fluid with afree surface on the rotating plane under the influence ofgravity and Coriolis force.Multi-layer generalizations possible.






Equations of motion

Dynamical variables: thickness of the layer h, horizontalvelocity v = (u, v), horizontal magnetic field b = (a,b).Equations of motion (Gilman, 2000):

∂tv + v · ∇v + f z ∧ v + g∇h =1h∇ (h b⊗ b) , (2.1)

∂th +∇ · (hv) = 0, (2.2)

∇ · (hb) = 0, (2.3)

∂tb + v · ∇b =1h∇ (h v⊗ b) , (2.4)

f = const > 0 - Coriolis parameter, and g - gravity acceleration.






1.5 dimensional RSWMHD

Under hypothesis of independence on x , written in componentsv = (u, v), b = (a,b), mRSW reads:

ut + vuy − fv = bay ,vt + vvy + fu = −ghy + bby ,

(2.5)

at + vay = buy ,bt + vby = bvy ,

(2.6)

ht + (hv)y = 0, (hb)y = 0. (2.7)

State of rest with any constant magnetic fieldb = B = (B1,B2) = const is a stationary solution.






Linear waves over the state of rest without meanmagnetic field, or with a zonal magnetic fieldB2 = const

Linearized system:

ut − f v = 0 ,vt + f u = −ghy ,

ht + Hvy = 0,(2.8)

at = bt = 0, Hby = 0. (2.9)

Dispersion relation for harmonic waves ∝ ei(ωt−ly):

ω = ±√

gHl2 + f 2. (2.10)

- inertia-gravity wavesV. Zeitlin Magneto-geostrophic adjustment





Linear waves over the state of rest with a meanmeridional magnetic field B1 = const

ut − f v = B2ay ,

vt + f u = −ghy + B2by ,

ht + Hvy = 0, Hby + B2hy = 0 ,at − B2uy = 0,bt − B2vy = 0.

(2.11)

Dispersion relation:

ω2 = (B2l)2 +gHl2 + f 2

2±

√(gHl2 + f 2

2

)2

+ f 2 (B2l)2, (2.12)

- mixed magneto-inertia-gravity waves.f → 0, B2 →∞ - Alfvèn waves.






Dispersion curves for two branches ofmageto-inertia-gravity waves

0.5 1.0 1.5 2.0 2.5 3.0

l

1

2

3

4

Ω





1.5D RSWMHDFrom 1.5D to full 2D RSWMHD system

Parameters and scaling

Scaling: U - velocity, B - magnetic field, L - coordinate y ,f−1 - fast time t , (εf )−1 - slow time T , H - thickness;Parameters: Rossby number Ro = U

f0L , magnetic Rossby

number Rom = Bf0L , Burger number Bu = gH

f 2L2 , nonlinearityparameter λ: h = H(1 + λη);Parameter regime: quasimagnetogeostrophy λBu

Ro = O(1),Ro ∼ Rom ∼ ε << 1.

Other choices:Ro Rom - small magnetic corrections to geostrophyRom Ro - magnetostrophy, imposes severe constraintson the leading-order configurations (Taylor states).






Nondimensional equations of motion

ut − v = −ε(

uT + vuy − Bay1+εη

),

vt + u + ηy = −ε(

vT + vvy + εB2ηy

(1+εη3

),

at = −ε(

aT + vay − Buy1+εη

),

ηt + vy = −ε(ηT + (vη)y

).

(3.1)

B - nondimensional analog of B, B 6= 0, otherwise the systemequivalent to 1.5D RSW with passive scalar a.






Multi-scale expansion

Two-time asymptotic expansion in ε:

(u, v , η,a) =(

u(0), v (0), η(0),a(0))

(x , y , t ,T )

+ ε(

u(1), v (1), η(1),a(1))

(x , y , t ,T ) +O(Ro2),

and eliminate resonances at each order by fast-time averaging,which allows to determine the slow-time dependence of thevariables.






Lowest order

u(0)t − v (0) = 0 ,

v (0)t + u(0) + η

(0)y = 0 ,

a(0)t = 0 ,

η(0)t + v (0)

y = 0.

(3.2)

Fast-time averaging→ Slow component: geostrophicallybalanced zonal flow plus zonal magnetic field

v (0) = 0, u(0) + η(0)y = 0, a(0) = a(0)(T , y). (3.3)

Fast component: non-dimensional version of the system (2.8),inertia-gravity waves.






First order

u(1)t − v (1) = −

(u(0)

T + v (0)u(0)y − Ba(0)

y

),

v (1)t + u(1) + η

(1)y = −

(v (0)

T + v (0)v (0)y

),

a(1)t = −

(a(0)

T + v (0)a(0)y − Bu(0)

y

),

η(1)t + v (1)

y = −(η(0)T +

(v (0)η(0)

)y

).

(3.4)






Slow motion

Fast-time averaging + (3.3)→ evolution of slow variables:

a(0)T + Bη(0)yy = 0,(

η(0) − η(0)yy

)T− Ba(0)

yy = 0.(3.5)

Solution in harmonic form with the wavenumber l andfrequency ω ⇒ dispersion:

ω2 = B2 l4

1 + l2. (3.6)

This is small Rossby-number limit of the lower branch of thedispersion relation of magneto-inertia-gravity waves . Slowcomponent: a packet of rotation-modified Alfvèn waves.






Initialization problem

Question:How to split arbitrary initial conditions into fast and slow parts?

(u, v , η,a)t=0 = (uI(y), vI(y), ηI(y),aI(y)) (3.7)

Answer:slow component follows (3.5) with initial conditions:

η(0)I =

(1− ∂2

yy

)−1(ηI + uIy ), aI = aI , (3.8)

fast component follows u0tt + u0 − u0yy = 0 with initialconditions

u0I = uI + ηIy , u0tI = vI . (3.9)






Staying close to 1.5D: disparity of spatial scales

Different scalings in x and y : δ =LyLx 1→

ut − v + ε(vuy + δuux ) + δηx = ε(δaax + bay ),vt + u + ε(vvy + δuvx )ηy = ε(δabx + bby ),

ηt + [((1 + εη) v ]y + δ [((1 + εη) u]x = 0at + ε(vay + δuax ) = ε(δaux + buy ),bt + ε(vby + δubx ) = ε(δavx + bvy ),

[((1 + εη) b]y + δ [((1 + εη) a]x = 0.(3.10)

Choice: δ ∼ ε.






Two-time asymptotic expansion in ε: lowest order

Same equations as in the 1.5D case,

b(0)y = 0, b(0)

t = 0.⇒ (3.11)

v (0) = 0, u(0) + η(0)y = 0, (3.12)

and both components of magnetic field are slow:a(0) = a(0)(x , y ,T ), b(0) = b(0)(x ,T ).






Two-time asymptotic expansion in ε: next order

u(1)t − v (1) = −

(u(0)

T + v (0)u(0)y + η

(0)x − b(0)a(0)

y

),

v (1)t + u(1) + η

(1)y = −

(v (0)

T + v (0)v (0)y

),

η(1)t + v (1)

y = −(η(0)T +

(η(0)v (0))

y + u(0)x

)v (1)

t = −(

a(0)T + v (0)a(0)

y − b(0)u(0)y

),

b(1)t = −

(b(0)

T − b(0)v (0)y

)b(1)

y = −(

a(0)x + b(0)η

(0)y

).

(3.13)






Equations for slow motion

−a(0)T + b(0)(x)η

(0)yy = 0,(

η(0) − η(0)yy

)T− b(0)(x)a(0)

yy = 0.(3.14)

As follows from the last equations in (3.13) b(0)T = 0. We thus

have the equation of slow Alfvèn waves with modulation in thetransverse to the magnetic field direction - perfect agreementwith 1.5D case.






Full 2D: δ = 1. Lowest order in ε

u(0)t − v (0) + η

(0)x = 0 ,

v (0)t + u(0) + η

(0)y = 0 ,

η(0)t + v (0)

y + u(0)x = 0 ,

a(0)t = 0 ,

b(0)t = 0 ,

a(0)x + b(0)

y = 0.

(3.15)

⇒ both components of the magnetic field are slow.






Equations of slow motion

−v (0) + η(0)x = 0 ,

+u(0) + η(0)y = 0 ,

v (0)y + u(0)

x = 0 ,a(0)

x + b(0)y = 0.

(3.16)

⇒ hydrodynamic slow component is in geostrophic balance,magnetic field is slow. Equations: diagnostic.






First order in ε

u(1)t − v (1) + η

(1)x = Ru,

v (1)t + u(1) + η

(1)y = Rv ,

η(1)t + v (1)

y + u(1)x = Rη,

v (1)t = Ra,

b(1)t = Rb,

v (1)x + b(1)

y = Rdiv .

(3.17)






Here

Ru = −(

u(0)T + u(0)u(0)

x + v (0)u(0)y + η

(0)x − a(0)a(0)

x − b(0)a(0)y

),

Rv = −(

v (0)T + u(0)u(0)

x + v (0)v (0)y − a(0)b(0)

x − b(0)b(0)y

),

Rη = −(η(0)T +

(η(0)u(0))

x +(η(0)v (0))

y

),

Ra = −(

a(0)T + u(0)a(0)

x + v (0)a(0)y − a(0)v (0)

x − b(0)u(0)y

),

Rb = −(

b(0)T + u(0)b(0)

x + v (0)b(0)y − a(0)v (0)

x − b(0)v (0)y

),

Rdiv = −((

a(0)η(0))

x +(b(0)η(0)

)y

).

(3.18)






Slow motions: magnetic field

a(0)T + u(0)a(0)

x + v (0)a(0)y − a(0)u(0)

x − b(0)u(0)y = 0,

b(0)T + u(0)b(0)

x + v (0)b(0)y − a(0)v (0)

x − b(0)v (0)y = 0.

(3.19)

Only slow components of velocity contribute. Constraint ofnon-divergence→ magnetic potential A(0):

a(0) = A(0)y , b(0) = −A(0)

x .⇒ (3.20)

A(0)T + J (η(0),A(0)) = 0, (3.21)

where J - Jacobian.






Slow motions: hydrodynamics

Combining equations in (3.17), we arrive to(v (1)

x − u(1)y − η(1)

)t

= Rη − (Rv )x + (Ru)y . (3.22)

Time-averaging, with the help of equations (3.16):(−η(0) +∇2η(0)

)T

+ J (η(0),∇2η(0))− J (A(0),∇2A(0)) = 0.(3.23)






Slow motions with external magnetic field

Magnetic field with a constant component in y -direction:A(0) = −Bx +A(0)

1 (x , y ,T )→

(A(0)1 )T + Bη(0)y + J (η(0),A(0)

1 ) = 0,(−η(0) +∇2η(0)

)T − B∇

2A(0)1 + J (η(0),∇2η(0))− J (A(0)

1 ,∇2A(0)1 ) = 0.

(3.24)In the absence of x-dependence these equations areequivalent to (3.14). No fast-motion drag upon the slowmotions!






Resumé of theoretical results:

Slow and fast motions: rotation-modified Alfvèn andmagneto-inertia-gravity waves. Dynamically split at small εMagnetostrophic adjustment: rapid evacuation ofinertia-gravity waves leaving Alfvèn waves. Slowcomponent propagative, all of the initial perturbationeventually dispersed.





Numerical scheme and setup:

CodeFinite-volume, well-balanced (with topography, magnetic fieldand rotation), resolving all characteristics (material, Alfvèn,magneto-gravity), entropy satisfying (Bouchut and Lhebrard,2014). Benchmarked with exact nonlinear waves. Magneticconstraint incompatible with hydrostatic reconstruction used forbalancing. Relaxed and checked aposteriori.

Setup

Unbalanced Gaussian zonal jet u(y) = Ue−y2

L2 with Ro = 0.1without (Rossby adjustment) or with (magnetostrophicadjustment) transverse magnetic field with Rom = Ro.





Rossby adjustment

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

-20 -15 -10 -5 0 5 10 15 20

h|t=2 h|t=6 h|t=10h|t=14

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-20 -15 -10 -5 0 5 10 15 20

u|t=2 u|t=6 u|t=10 u|t=14

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-20 -15 -10 -5 0 5 10 15 20

v|t=2 v|t=6 v|t=10v|t=14





Magnetostrophic adjustment: hydrodynamics

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

-20 -15 -10 -5 0 5 10 15 20

h|t=2 h|t=6 h|t=10h|t=14

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-20 -15 -10 -5 0 5 10 15 20

u|t=2 u|t=6 u|t=10 u|t=14

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-20 -15 -10 -5 0 5 10 15 20

v|t=2 v|t=6 v|t=10v|t=14





Magnetostrophic adjustment: magnetics

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-20 -15 -10 -5 0 5 10 15 20

a|t=2 a|t=6 a|t=10a|t=14

0.16

0.165

0.17

0.175

0.18

0.185

0.19

-20 -15 -10 -5 0 5 10 15 20

b|t=2 b|t=6 b|t=10b|t=14





Conclusions and discussion

Fast-slow dynamical splitting operational in hydrostaticMHD with fast rotationMagneto-geostrophic adjustment: evacuation of fastinertia-gravity waves, rotation-modified Alfvèn waves - partof slow motion.QG MHD models heuristically used in the literature areconsistentCrucial role of rotation-modified Alfvèn waves -confirmation of relevance of Alfvèn-wave turbulenceMulti-layer/continuos stratification generalizations withinclusion of baroclinic effects straightforward.





Published in:

V. Zeitlin, Ch. Lusso and F. Bouchut, "Geostrophic vsmagneto-geostrophic adjustment and nonlinearmagneto-gravity waves in rotating shallow watermagnetohydrodynamics" GAFD, 2015


Documents

Magneto-geostrophic adjustment in rotating shallow water ......2015/09/21 · This is small Rossby-number limit of the lower branch of the dispersion relation of magneto-inertia-gravity