theoretical and numerical studies of ionospheric...

Theoretical and numerical studies

of ionospheric irregularities

Tatsuhiro Yokoyama

National Institute of Information and

Communications Technology (NICT), Japan

Outline

• Physics-based models

• Empirical models

• Numerical techniques

• Rayleigh-Taylor instability

• Perkins instability and E-F coupling

• Summary

Physics-based Models

Global Meteorological Simulation

From Japan Meteorological Agency website

What is Numerical Simulation?

Reproduce space environment virtually

in the computational space

• Density

• Velocity

• Temperature

• Electric fields

Computer

Ionosphere model

What is Numerical Simulation?

Inflow and outflow of

state variables

＋－

Ionization due to

solar radiation

Chemical reactions

O+

NO+

O2+

e-

O

N2

O2

Electric field

Geomagnetic

field lines

Effects of

electromagnetic

force

Collision with

neutral particles

State variables are updated every time step based on first-principles equations.

Equations

vit

vi vi 1

minip in vi vn g

e

miE vi B

3

2nik

Tit

qi t

nimi itmi mt

3k Tt Ti mt vi vt 2

3

2nek

Tet

qe Qe

nit

nivi Pi Li From atmospheric model

From electrodynamics model Photoionization&

Chemical reactions

Ion-neutral collisions Lorentz force

Heat flux Effect of collisions

Photoelectron heating Heat flux

Ion density

Ion velocity

Ion temperature

Electron temperature

Ionospheric Model Results

O+density Electric field

Time evolution of state variables at 400 km height

Electron temperature

Ground-to-topside model of Atmosphere and Ionosphere for Aeronomy（GAIA）

Ionosphere model

-Shinagawa and Oyama [EPS, 2006]

-Physics and chemistry for

ions (N2+, O2

+, NO+, O+ , H+)

-Grid size: 1°×5°×10-100 km

-Height range: ~100 to 3000 km

Whole atmosphere GCM

-Miyoshi and Fujiwara [GRL, 2003]

-physics and chemistry

from troposphere to thermosphere

T, u, v, w, N(O), N(O2), …

-Grid size: 2.8°×2.8°× 0.4H

Electrodynamics model

-Jin et al. [JGR, 2008]

-Solve current continuity with J=σ・（E+U×B） for electrostatic potential

-Equipotential magnetic field line

Atmosphere-Ionosphere Coupling

Rain fall rate

on ground

Electron density

distribution at 400

km height

condensation of

water vapor

→ latent heat release

upward propagation of

atmospheric waves

[Jin et al., 2011]

“Wave-4” observed by

IMAGE satellite

[Immel et al., 2006]

Available Models for Community

SAMI2 --- Ionosphere model developed by NRL (Naval Research

Laboratory, USA), 2-dimensional on the same meridional plane.

http://www.nrl.navy.mil/ppd/branches/6790/sami2

TIE-GCM --- thermosphere-Ionosphere-electrodynamics model

developed by NCAR (National center of atmospheric research, USA),

community model

http://www.hao.ucar.edu/modeling/tgcm/

GAIA --- whole atmosphere-ionosphere coupled model developed by

Kyushu Univ, Seikei Univ. and NICT, model itself not available currently

(model size is too large) but the result data is available

http://seg-web.nict.go.jp/GAIA/index_e.html

SAMI2

Model Comparison

Fang et al. (2013)

Models are not perfect.

There are large discrepancies

between various models.

Empirical Models

Empirical Models

ρ

x/ｔ（position/time）

ρ（x） = a x^3 + b x^2 + c x + d

observation

data

• Described by arbitrary fitting functions with coefficients

determined statistically by observation data

• The coefficients are dependent on locations, time, and

physical variables, and usually functions of such input

parameters as F10.7, Ap index,…

• examples: IRI(ionosphere), MSIS(neutral), HWM(neutral

wind), EXB drift,…

True profile

lack of observation

IRI (International Reference Ionosphere)

Input Year, Day of year, UT, Altitude, Geodetic latitude and

longitude, Sunspot number (Rz12), and Ionospheric index

(IG12).

Output Electron density, electron temperature, ion temperature,

ion composition (O+, H+, He+, NO+, O2+), ion drift, TEC

Cover Regions global, 50km-2000km

Conditions Monthly averages in the non-auroral ionosphere for

magnetically quiet conditions

Data Sources worldwide network of ionosondes, the incoherent scatter

radars (Jicamarca, Arecibo, Millstone Hill, Malvern, St.

Santin), the ISIS and Alouette topside sounders, and in

situ instruments on several satellites and rockets

Comparison Between GAIA, IRI and Ionosonde

IRI GAIA

ionosonde

(Kokubunji)

Solar maximum Solar minimum

Comparison Between GAIA, IRI and Ionosonde

GAIA, NmF2 IRI, NmF2

IRI

GAIA

ionosonde

(Wakkanai, Kokubunji,

Yamagawa, Okinawa)

For monthly average,

IRI shows very good results.

MSIS (Mass Spectrometer Incoherent Scatter)


longitude, F10.7 index (for previous day and three-month

average), and Ap index (daily or Ap history for the last 59

hours).

Output Neutral densities (He, O, N2, O2, Ar, H, and N, total mass

density) and temperature


Data Sources Several rockets, satellites (OGO 6, San Marco 3, AEROS-A,

AE-C, AE-D, AE-E, ESRO 4, and DE 2), and ISR (Millstone

Hill, St. Santin, Arecibo, Jicamarca, and Malvern) for

thermosphere.

Below 72.5 km, zonal average temperature and based on

the MAP Handbook tabulation. Below 20 km,

supplemented with averages from the National

Meteorological Center (NMC).

Comparison Between GAIA and MSIS

MSIS

GAIA

Empirical model and physics model show good agreement

at 300km but not at 110 km. There is a lack of observation

in the lower thermosphere.


MSIS GAIA

Neutral temperature at 300 km height

GAIA, Tn MSIS, Tn


MSIS

GAIA

neutral temperature at 110 km height

GAIA, Tn MSIS, Tn

They are different in mean value,

wave amplitude and phase…

HWM (Horizontal Wind Model)


longitude, F10.7 index (for previous day and three-month

average), and Ap index (daily or Ap history for the last 59

hours).

Output Neutral wind velocities (zonal and meridional

components)


Data Sources Wind data obtained from the AE-E and DE 2 satellites,

incoherent scatter radar and Fabry-Perot optical

interferometers, and MF/Meteor data

Comparison Between GAIA and HWM

HWM GAIA

zonal wind at 300 km height

GAIA, Un HWM, Un

They are in good agreement

at least at equator.

Comparison Between GAIA and HWM

HWM GAIA

zonal wind at 110 km height

GAIA, Un HWM, Un

They look very different ,

including tidal variations.

Comparison Between Empirical and Physics-Based Model

Empirical models Physics-based models

Approach Statistical approach, fitting

to arbitrary functions

Solving the physics equations

(energy, momentum…)

Prediction

capability

Climatological behaviors

(location, seasonal, solar

cycle dependences, ..),

quiet time behaviors

day-to-day variations,

disturbances due to magnetic

storms and lower atmospheric

inputs

Accuracy good where sufficient

observation data exists, and

bad for others (e.g., lower

thermosphere)

Good for some regions and

disturbances, but generally

difficult to make good agreement

with observations due to

problems of numerical treatments

usage Reference use, prediction of

climatology, input for

regional physics-based

models, etc.

prediction, analysis of physics,

numerical experiments, etc.

Computationa

l requirements

Not heavy Depends on the model and

resolution, usually heavy

Numerical Technique

How to Model?

Finite difference method

Finite volume method

Finite element method

…

i-1, i, i+1

i-2 i-1 i i+1

Initial Value Problem

0

x

cv

t

cCentral Difference 0

2

11

1

x

ccv

t

cc n

i

n

i

n

i

n

i

)(2

11

1 n

i

n

i

n

i

n

i ccx

tvcc

i-1, i, i+1, i+2

n+1

n

x

t

Central Difference Scheme

Unconditionally unstable!

Initial Value Problem

0

x

cv

t

cUpwind Difference 01

1

x

ccv

t

cc n

i

n

i

n

i

n

i

)( 1

1 n

i

n

i

n

i

n

i ccx

tvcc

(if v > 0)

i-1, i, i+1, i+2

n+1

n

x

t

v

Upwind Difference Scheme

Numerical diffusion is a serious problem.

Various Algorithm for Transport Equation

CIP Scheme

Constrained Interpolation Profile (CIP) method

Advection of density gradient as dependent variable

Yabe et al. (2001)

CIP Scheme

No diffusion, no ripple.

Boundary Value Problem

0 J b 2

ji

jijijijijijib

yx,2

1,,1,

2

,1,,1 22

i-1, i, i+1 x

j+1

j

y

j-1

bAx

How to Solve Ax=b?

Direct solution (e.g., Gaussian elimination)

– An order of N3 operation. Non-zero elements are filled-in.

⇒ Suitable for dense matrix

Iterative methods (Jacobi, Gauss-Seidel, SOR)

– Matrix-Vector multiplication at each iteration

⇒ Suitable for sparse matrix

Multigrid method

bAx ULDA

))( n11nxUL(bDx

Conjugater Gradient (CG) Method

Non-stationary iterative method, while SOR-type

is called stationary iterative method.

Green: steepest descent

Red: conjugate gradient

Various Algorithms are still invented

van der Vorst (1992)

Fujino and Murakami (2013)

Convergence of Iterative Methods

Zhang (1997)

The best algorithm is problem-dependent.

High Performance Computing

Early supercomputer consists of vector processor.

Vector-type machine becomes expensive compared

to parallel scalar-type machine.

Graphical Processing Unit (GPU) is also used to

accelerate simulation.

Huge vector processor Parallel scalar processor

Amdahl's Law

Writing efficient source code is critical.

S: SpeedUp

P: Parallelized portion

N: Number of processors

Rayleigh-Taylor Instability

Rayleigh-Taylor Instability

Unstable

Stable

Altitu

de

∇n

∇n

Linear Analysis of Rayleigh—Taylor Instability

𝜕𝑁

𝜕𝑡+ 𝛻 𝑁𝑽𝑖 = 0, 𝛻 ⋅ 𝑱 = 𝛻 ⋅ 𝑒𝑁 𝑽𝑖 − 𝑽𝑒 = 0

Incompressible plasma in the F region（𝛻 ⋅ 𝑽 = 0）

Zeroth-order vertical gradient only 𝜕𝑁

𝜕𝑧

Magnetic field diercts northward 𝐵 = 𝐵𝑦

Ignore second-order perturbation

𝜕𝑁

𝑑𝑡+

𝑀𝑔

𝑒𝐵

𝜕𝑁

𝜕𝑥−

1

𝐵

𝜕𝜙

𝜕𝑥

𝜕𝑁

𝜕𝑧= 0

𝜕𝑁

𝜕𝑥−

𝜈𝑖𝑛

𝑔𝐵𝑁

𝜕2𝜙

𝜕𝑥2= 0


Plane wave assumption for density and electrostatic

potential

𝜙 = 𝛿𝜙𝑒𝑖(𝜔𝑡−𝑘𝑥), 𝑁 = 𝑁0 𝑧 + 𝛿𝑁𝑒𝑖(𝜔𝑡−𝑘𝑥)

𝑖𝜔 − 𝑖𝑘𝑀𝑔

𝑒𝐵𝛿𝑁 +

𝑖𝑘

𝐵

𝜕𝑁0

𝜕𝑧𝛿𝜙 = 0

−𝑖𝑘𝛿𝑁 +𝜈𝑖𝑛

𝑔𝐵𝑁0𝑘2𝛿𝜙 = 0

Set determinant to be zero for coefficients of 𝛿𝑁, 𝛿𝜙,

resulting dispersion relation

𝜔 = 𝑘𝑀𝑔

𝑒𝐵− 𝑖

𝑔

𝜈𝑖𝑛

1

𝑁0

𝜕𝑁

𝜕𝑧


When the imaginary part of 𝜔 is negative (upward

density gradient), 𝛾 in 𝑒𝑖𝜔𝑡 = 𝑒𝑖𝜔𝑟𝑡𝑒𝛾𝑡 becomes

positive and the perturbation grows with time. 𝛾 is

called the linear growth rate.

𝛾 =𝑔

𝜈𝑖𝑛

1

𝑁0

𝜕𝑁

𝜕𝑧=

𝑔

𝜈𝑖𝑛𝐿

Preferable condition for the instability growth:

– Small collision frequency -> high altitude -> eastward

electric field

– Steep vertical density gradien -> recombination in the

bottomside around sunset time

Need to consider whole magnetic flux tube

– Simultaneous sunset at conjugate E regions -> sunset

terminator is parallel to magnetic declination

Growth Rate Estimation by Global Simulation

Global model does not

have enough spatial

resolution to reproduce

plasma bubbles.

Using output parameters,

the growth rate of the

Reyleigh-Taylor instability

can be estimated.

General seasonal and

longitudinal pattern can

be well explained.

Wu (2015)

History of Plasma Bubble Modeling


In the review paper by Woodman (2009), “…The qualitative theory of

Woodman and La Hoz (1976) was soon supported by numerical

simulations (Scannapieco and Ossakow, 1976) (this sequence of events is

sometimes inverted in review papers and historical introductions). The idea

of a bubble ‘floating’ to the top was first presented by Woodman at the

1975 Gordon Conference on Space Plasma Physics where it was suggested

that the NRL code used to explain striations in Barium cloud releases could

simulate the bubble formation…”

Woodman and LaHoz (1976)

Received Jan. 26, 1976, published Nov. 1976

Scannapieco and Ossakow (1976)

(Received Apr. 5, 1976, published Aug. 1976)


Zalesak and Ossakow (1980)

Three Layer Model

Zalesak et al. (1982)

New Numerical Scheme

Huba and Joyce (2007)

First Three-Dimensional Model

Keskinen et al. (2003)

SAMI3/ESF Model

Huba et al. (2008) Based on global SAMI3 model

SAMI3/ESF Model

Huba et al. (2010)

Cornell Model

Aveiro et al. (2012)

High Resolution Bubble (HIRB) Model by NICT

Dipole orthogonal coordinate

Maximum spatial resolution perpendicular to B is 200 m.

(NX(B||), NY(BL), NZ(B)) = (501, 3600, 1680)

O+ (F region), NO+ (E region)

Yokoyama et al. (2014)

Highest Resolution Model (200m x 200m)

Perkins Instability and E-F Coupling

Medium-Scale Traveling Ionospheric Disturbances (MSTID)

630-nm airglow GPS-TEC

3m-scale irregularities

（MU radar）

Saito et al. (2001)

Conjugate MSTID in Northern/Southern Hemisphere

Otsuka et al. (2004)

Perkins Instability

Perkins instability can produce banded structure.

Linear growth rate is very small.

Field line-integrated model for theoretical and

numerical studies (e.g., Kelley and Miller, 1997).

E-F coupling is important (e.g., Cosgrove et al., 2004).

0

0

Perkins (1973)

Zhou and Mathews (2006)

Simulation of Perkins Instability

First 2D simulation of Perkins instability in 1997.

Tilted bands are produced, but the growth rate is too small.

Kelley and Miller (1997)

Sporadic E (Es) Layer

Long-lived metallic ions survive even after sunset.

They are gathered by neutral wind shears and form

thin layers.

Mathews et al. (1997)

Perkins-Type Instability in the E Region

Similar mechanism also works in the E region.

Cosgrove and Tsunoda (2002)

Tsunoda et al. (2004)

Hysell et al. (2004)

Basic Idea of E-F Coupling

Westward wind on Es patch generates eastward

polarization electric field (upward ExB drift in the F region).

Haldoupis et al. (2003)

E-F Coupling Evidence

Irregularities occur simultaneously in both regions

along the same magnetic field.

F region echoes （projected on 100km）

E region echoes

F region

E region

Sakata

MU radar

Alt

itu

de

Growth Rates of Coupled Instability

The second terms of (3) and (4) come from polarization

electric field mapped from the other region.

Pc ≫ P and E

c E.

Tsunoda (2006)

Simulation Model for E-F Coupling Instability

O+ (F region), NO+ (E region),

Fe+ (Es layer), and electrons.

45⁰ inclination of B.

The altitude range is 90 -

470km. The grid spacing is

2km in each direction in F

region and 500m in the E

region.

Periodic boundary in

horizontal directions. Yokoyama et al. (2009)

E-F Coupling Simulation

Time variation on a meridional plane.

The F-peak altitude is modulated 8km, and the

conductivity variation reaches more than 30%.

Growth Rate

F-region conductivity variation reaches 10% after 700s,

while it takes 7000s by the isolated Perkins instability.

New Model with Dipole Magnetic Field

Random perturbation + zonal wind shear in the E region

Yokoyama and Hysell (2010)

Southward Propagation by E Region Neutral Wind

E region F region

Random perturbation + rotational shear in the E region

E

S

W

Es

Scale Dependence of MSTID

Key results:

Cases 3 and 4 grew most rapidly

Scale distribution of MSTIDs converged to 100-200 km

Very long frontal structures of MSTIDs were formed.

Case 1 2 3 4 5

Perturbation

scale 20km 40km 80km 160km 320km

E-region wind Zonal wind shear (60 m/s)

F-region wind Southwestward (120 m/s)


E region

F region

~40 km scale ~160 km scale

Yokoyama (2013)


E region

F region

Growth Rate Wavelength

Case 1 2 3 4 5

Scale

(km) 20 40 80 160 320

Polarization in Es Layer

Smaller-scale Es perturbation:

produces larger Ep by larger .

is reformed by wind shear more quickly.

small

small Ep

Large

Large Ep

Yokoyama et al. (2009)

Long Frontal Formation

Polarization process along the

wavefront makes it uniform due to

Perkins “stability” mechanism.

It does not prevent Perkins “instability”

because Jp can flow uniformly along it.

E F

Suggested References

• Global model

Fang, T.-W. et al., Comparative studies of theoretical models in the

equatorial ionosphere, Modeling the Ionosphere-Thermosphere System,

Geophysical Monograph Series, 133-144, AGU, 2013.

• Rayleigh-Taylor instability

Sultan, P. J., Linear theory and modeling of the Rayleigh-Taylor instability

leading to the occurrence of equatorial spread F, J. Geophys. Res., 101,

26,875-26,891, 1996.

Yokoyama, T., A review on the numerical simulation of equatorial plasma

bubbles toward scintillation evaluation and forecasting, Prog. Earth

Planet. Sci., 4:37, 2017.

• Perkins instability

Tsunoda, R. T., On the coupling of layer instabilities in the nighttime

midlatitude ionosphere, J. Geophys. Res., 111, A11304, 2006.

Yokoyama, T. et al., Three-dimensional simulation of the coupled Perkins

and Es-layer instabilities in the nighttime midlatitude ionosphere, J.

Geophys. Res., 114, A03308, 2009.

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