Variability of Stochastically Forced Zonal Jets Laura Cope , Peter - - PowerPoint PPT Presentation

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Variability of Stochastically Forced Zonal Jets Laura Cope , Peter Haynes Department of Applied Mathematics and Theoretical Physics, University of Cambridge MOTIVATION Gaseous Giant Planets Earths Atmosphere & Oceans What insights can

SLIDE 1

Variability of Stochastically Forced Zonal Jets

Laura Cope, Peter Haynes

Department of Applied Mathematics and Theoretical Physics, University of Cambridge

SLIDE 2

MOTIVATION

Gaseous Giant Planets Earth’s Atmosphere & Oceans

Credit: NASA/JPL-Caltech/Space Science Institute Credit: https://www.nasa.gov Credit: NASA / Science Photo Library Credit: NASA, ESA

What insights can we learn about the variability of jet streams using idealized models?

SLIDE 3

OBSERVATIONS

Earth’s Atmosphere Earth’s Oceans

Increasing time variability

(earth.nullschool.net) (Sokolov, Rintoul 1996) 40°S 50°S 60°S 1995 1996 Year

Jupiter

Voyager 1979-1980

SLIDE 4

OBSERVATIONS

Earth’s Atmosphere Earth’s Oceans

Increasing time variability

(earth.nullschool.net) (Sokolov, Rintoul 1996) 40°S 50°S 60°S 1995 1996 Year

Jupiter

Voyager 1979-1980 Cassini 2000

SLIDE 5

40°S 50°S 60°S 40°S 50°S 60°S 1995 1996 Year

SLIDE 6

OVERVIEW OF IDEALIZED MODELS

Planetary rotation Turbulence Jets Friction

SLIDE 7

IDEALIZED MODELS: MATHEMATICAL FORMULATION

Stochastic Force

Fourier space: Physical space:

Equation of Motion: Vorticity Equation

Stochastic forcing Linear friction Hyper- viscosity Planetary Rotation Beta, Energy input rate,. Damping rate, .

µ

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2D (barotropic) Doubly periodic Beta plane Stochastic force

Features

𝜖𝜂 𝜖𝑢 + 𝒗 & ∇𝜂 + 𝛾𝑤 = 𝜊 − 𝜈𝜂 + 𝜉/∇0/𝜂

SLIDE 8

Generalized Quasilinear Approximation Marston, Chini, Tobias (2016) Reference: Increasing zonal wavenumber

IDEALIZED MODELS: MATHEMATICAL FORMULATION

SLIDE 9

Generalized Quasilinear Approximation Marston, Chini, Tobias (2016) Reference: Low modes ≤ 𝚳 High modes > 𝚳 Increasing zonal wavenumber Separation = 𝚳

IDEALIZED MODELS: MATHEMATICAL FORMULATION

SLIDE 10

Generalized Quasilinear Approximation Marston, Chini, Tobias (2016) Reference: Low modes ≤ 𝚳 Increasing zonal wavenumber High modes > 𝚳 Separation = 𝚳

IDEALIZED MODELS: MATHEMATICAL FORMULATION

SLIDE 11

Overview

Low modes ≤ 𝚳 High modes > 𝚳 Generalized Quasilinear Approximation

Low-high mode decomposition: Basic vorticity equation: Low modes: High modes: Vorticity equation:

𝜔 = 5

|7|89

𝑓;7< = 𝜔7 + 5

|7|>9

𝑓;7< = 𝜔7 = ? 𝜔 + 𝜔′

AB AC = ℒ 𝜂 + 𝒪[𝜂, 𝜂] A? B AC = ℒ

̅ 𝜂 + H 𝒪[ ̅ 𝜂, ̅ 𝜂] + H 𝒪[𝜂J, 𝜂J]

ABL AC = ℒ 𝜂J + 𝒪J[ ̅

𝜂, 𝜂J] +[HHNL]

𝜖𝜂 𝜖𝑢 + 𝒗 & ∇𝜂 + 𝛾𝑤 − HHNL = 𝜊 − 𝜈𝜂 + 𝜉/∇0/𝜂

IDEALIZED MODELS: MATHEMATICAL FORMULATION

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SUMMARY OF IDEALIZED MODELS

Reduction in nonlinearity

Nonlinear (Λ = 𝑂) Generalized Quasilinear (Λ = 1) Quasilinear (Λ = 0)

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NUMERICAL SIMULATIONS – NONLINEAR (NL) MODEL

Zonal velocity field Zonal mean zonal velocity Zonal mean zonal velocity evolution in time

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NONLINEAR (NL) MODEL - TYPES OF VARIABILITY

Randomly wandering Merging & nucleating Migrating

New type of variability found: jets migrate north and south

Result 1

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ALL MODELS - TYPES OF VARIABILITY

Randomly wandering Merging & nucleating Migrating QL Model (Λ = 0) No clear migration GQL Model (Λ = 1) NL Model (Λ = 𝑂)

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A CLOSER LOOK AT ZONAL JET MIGRATION

Question 1 Question 3 Question 2

Why do jets migrate only when Λ ≥ 1?

Low modes ≤ 𝚳 High modes > 𝚳

Do jets migrate in more complex systems? Can we predict the speed

f migration?

SLIDE 17

A CLOSER LOOK AT ZONAL JET MIGRATION

Question 1

Why do jets migrate only when Λ ≥ 1?

Low modes ≤ 𝚳 High modes > 𝚳

Do jets migrate in more complex systems? Can we predict the speed

f migration?

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AN INTRODUCTION TO ZONONS

Reference: Sukoriansky, Dikovskaya, Galperin (2008), PRL

NL Model (Λ = 𝑂) Zonons (Nonlinear waves)

Coherent structures excited by Rossby waves with same 𝑙< and same phase speed Nonlinear zonons Linear Rossby wave

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AN INTRODUCTION TO ZONONS

Reference: Sukoriansky, Dikovskaya, Galperin (2008), PRL

Zonons (Nonlinear waves)

Coherent structures excited by Rossby waves with same 𝑙< and same phase speed Nonlinear zonons Linear Rossby wave

GQL Model (Λ = 1)

SLIDE 20

AN INTRODUCTION TO ZONONS

Reference: Sukoriansky, Dikovskaya, Galperin (2008), PRL

Zonons (Nonlinear waves)

Coherent structures excited by Rossby waves with same 𝑙< and same phase speed Nonlinear zonons Linear Rossby wave

QL Model (Λ = 0)

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NL Model QL Model Not Migrating Migrating North No clear migration

Migration requires Λ ≥ 1 when jets and zonons coexist

Result 2

jet zonons

Q1: WHY DO JETS MIGRATE ONLY WHEN 𝚳 ≥ 𝟐?

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NL Model QL Model Not Migrating Migrating North No clear migration

Migration requires Λ ≥ 1 when jets and zonons coexist

Result 2

jet zonons

Q1: WHY DO JETS MIGRATE ONLY WHEN 𝚳 ≥ 𝟐?

SLIDE 23

Migrating north Migrating south Not migrating 𝜂(𝑦, 𝑧, 𝑢) Schematic

Q1: WHY DO JETS MIGRATE ONLY WHEN 𝚳 ≥ 𝟐?

SLIDE 24

Migrating north Migrating south Not migrating 𝜂(𝑦, 𝑧, 𝑢) Schematic

Q1: WHY DO JETS MIGRATE ONLY WHEN 𝚳 ≥ 𝟐?

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Result 3

Migration requires an asymmetric eddy forcing and mean flow

Reynolds stress force Zonal mean flow

Zonal mean forcing

AH [ AC = 𝑤′𝜂′ −𝜈?

𝑣

Q1: WHY DO JETS MIGRATE ONLY WHEN 𝚳 ≥ 𝟐?

SLIDE 26

A CLOSER LOOK AT ZONAL JET MIGRATION

Question 2

Why do jets migrate only when Λ ≥ 1? Do jets migrate in more complex systems? Can we predict the speed

f migration?

SLIDE 27

Q2: CAN WE PREDICT THE SPEED OF MIGRATION?

ZMF index Jet migration speed

0.3 0.9

Result 4

Migration speed is given approximately by: speed ∝ (ZMF)ef

Zonal mean flow index (ZMF)

ZMF = Fraction of energy in mean flow 1

No jets No eddies Eddies & jets

𝜀𝑦 𝜀𝑢 speed = 𝜀𝑦 𝜀𝑢

SLIDE 28

Result 4

Zonal mean flow index (ZMF)

ZMF = Fraction of energy in mean flow 1

No jets No eddies Eddies & jets

speed = 𝜀𝑦 𝜀𝑢

log (Jet migration speed) slope = -3 log (ZMF index)

Migration speed is given approximately by: speed ∝ (ZMF)ef

𝜀𝑦 𝜀𝑢

Q2: CAN WE PREDICT THE SPEED OF MIGRATION?

SLIDE 29

A CLOSER LOOK AT ZONAL JET MIGRATION

Question 3

Why do jets migrate only when Λ ≥ 1? Can we predict the speed

f migration?

Do jets migrate in more complex systems?

SLIDE 30

Q3: DO JETS MIGRATE IN MORE COMPLEX SYSTEMS?

General circulation model: Poleward drift Atmosphere observations: Poleward drift General circulation model: Equatorward drift

30°S 60°S 90°S EQ 150 days Kemke & Kaspi (2015), JAMES Feldstein (1998), JAS Young, Read & Wang (2019), Icarus

SLIDE 31

CONCLUSIONS

OBJECTIVE METHOD MODELS

Study of jet stream variability Rotation + turbulence + friction = zonal jet Generalized quasilinear approximation

SLIDE 32

CONCLUSIONS

Migration requires an asymmetric eddy forcing

4 2 1 3

Migration speed is approximately: speed ∝ (ZMF)ef New type of jet variability found: migrating jets Migration requires Λ ≥ 1 when jets and zonons coexist