Running the code with different initial polar fields

The code was run with different initial polar fields.

1. Initial field of 10 Gauss

OUTPUT:


Peak initial field for the 3rd cycle is -7.55G in the northern hem which occurs at 57.1233 degree lat.@25.3yrs.
The lowest that occurs on the same latitude is -0.86G.@14.2yrs.
Difference=6.7G

So, the cycles go like:

    Year                         Polar field (Gauss)
    14.2                               -0.86
    25.3                               -7.55
    36.4                               -0.86
    47.5                               -7.55


2. Initial field of 9 Gauss

OUTPUT:


Peak initial field for 3rd cycle is -6.7827G in the northern hem which occurs at 57.1233 degree lat.@25.2133yrs.
The lowest that occurs on the same latitude is -0.10699G.@14.1321yrs. and -0.15G.@35.8679yrs.
Difference=6.68G

So, the cycle goes like:

  Year                         Polar field (Gauss)

    14.1                               -0.11

    25.2                               -6.78

    35.9                               -0.15

    47.5                               -6.82

3. Initial field of 8 Gauss

OUTPUT:

Peak initial field for 3rd cycle is -6.06 G in the northern hem which occurs at 57.1233 degree lat.@24.9yrs.

The lowest that occurs on the same latitude is +0.61 G.@14.1yrs. and -0.61G.@35.9yrs.

Difference=6.67G

So, the cycle goes like:

  Year                         Polar field (Gauss)

    14.1                               +0.61

    24.9                               -6.06

    35.9                               +0.61

    46.7                               -6.06

4. Initial field of 7 Gauss:

OUTPUT:

  Year                         Polar field (Gauss)

    14.1                               +1.41

    25.3                               -5.26

    35.9                               +1.38

    47.3                               -5.28

    58.0                              +1.36

    68.8                               -5.31

Difference=6.67G

5. Initial field of 6 Gauss:

OUTPUT:

Year                         Polar field (Gauss)

    14.1                               +2.16

    25.3                               -4.5

    35.9                               +2.16

    47.3                               -4.5

Difference=6.66G


For realistic result, the difference should be around 20 Gauss.

The input parameters that can be varied to achieve this are:
(1) No. of sunspots per cycle (These were modeled vaguely and hence remain doubtful).
(2) Separation between sunspots within a BMR(taken to be a constant=2R)
(3) The longitudes of eruption are totally random. The degree of randomness can be optimized.

Running the code with decreased randomization for calibration

To relatively stabilize the peak polar magnetic field, the input of one sunspot cycle was repeatedly fed into the surface flux transport code with changing polarity of magnetic field. The initial polar field is 4.5 Gauss above 60 degrees latitude.

The butterfly diagram of the simulation is :

The peak polar magnetic field values are:

Cycle          Peak polar magnetic field in Northern Hem        Peak polar magnetic field in Southern Hem
   1                                 +3.30                                                                         -3.15
   2                                  -3.35                                                                        +3.39
   3                                 +3.30                                                                         -3.15
   4                                  -3.35                                                                        +3.39
   5                                 +3.30                                                                         -3.15
   6                                  -3.35                                                                        +3.39
   7                                 +3.30                                                                         -3.15

The next step: Modelling a realistic input based on observations

For realistic solar cycle simulations, corresponding inputs have to be modeled and fed to the Surface flux transport code.

The modelling was done with reference to the study carried out by Jie Jiang et al. (http://arxiv.org/abs/1102.1266v1) and van Ballegooijen et al. 1998.

The input parameters needed are:

(1) Time of BMR eruption

(2) Latitude of eruption

(3) Tilt angle (with respect to the solar equator)

(4) Longitude of eruption

(5) Radius of individual spots in a BMR

(6) Peak magnetic field (Bmax)

(7) Separation between the centers of the individual spots.

One solar cycle of 11 years was divided in 120 phases. And the sunspots were placed at their respective locations after each phase. The following figure shows the latitude of eruption(in degrees) vs phase that was fed into the surface flux transport code:

An initial polar field of +-4.5 Gauss was placed within 23 degrees of the poles, and the surface flux transport code was given a run for 12 sunspot cycles(132 years). This is what was the output:

This is again a butterfly diagram, but you can really see butterfly like structures in it. The polar magnetic field reversal is clearly evident from the diagram.

But, the problem here is, the magnitude of the peak polar field is not constant. It varies from cycle to cycle.   

No. of Cycle     Peak Magnetic field in the northern hemisphere

        1                                             -4.5

        2                                            +3.8

        3                                            -2.8

        4                                            +5.0

        5                                            -3.8

        6                                            +3.4

        7                                            -3.9

        8                                            +2.7

        9                                            -3.9

       10                                           +3.9

       11                                           -4.9

       12                                           +2.3

       13                                           -5.0

Work up till now: Polar magnetic field reversal

An initial polar field of +-10 Gauss was placed within 20 degrees of the poles, and a BMR was placed in each hemisphere, such that the trailing spot has magnetic field of a polarity opposite to the polar magnetic field in that hemisphere. This automatically set the leading spots of the BMR to be opposite in polarity of magnetic flux.

Such BMRs were placed at regular intervals till the polar magnetic field was totally cancelled, and then became opposite.

The following figure is a butterfly diagram of the simulation. There is nothing like a butterfly in it, it is named like that for a different reason.

On the X-axis is the no. of time steps,

on the Y-axis is the latitude of the surface of the Sun,

and the colors indicate the value of magnetic field on a particular latitude at a particular time(averaged over all longitudes).

Work up till now: Building up of poloidal field

In this video, you can see,
When 2 BMRs in different hemispheres are such that, their leading spots have magnetic flux of the opposite sign, then their flux gets cancelled over the equator. While, the flux of the trailing sunspots is carried to the poles by the meridional flow.

But when the tilt of the BMRs is such that the leading spots have magnetic flux of the same sign, there is no cancellation over the equator, and both the poles end up getting the same polarity of magnetic field.

Work up till now

I have written the Surface flux transport code in FORTRAN77.

It seems to be working fine when I load an input with a single Bipolar Magnetic Region(BMR) placed near the equator. The diffusion seems to blow up the sunspots in size, and the meridional flow(11m/s) carries the poloidal component of the flux to the poles.

The above figures are contour plots of the radial magnetic field on the surface of the Sun. The X-axis is the azimuthal angle(phi) and the Y-axis is the polar angle(theta).

Purpose

I was introduced to research in astrophysics in the summer of 2011 when I went to IISER Kolkata for a summer project in Solar Physics, under the guidance of Dr. Dibyendu Nandi. He is one of the nicest person I have known.

My project was to develop a Surface Magnetic Flux Transport code that would simulate the evolution of radial magnetic field on the surface of the Sun.


Dr. Nandi, has himself worked on Solar Dynamo code. On combining the Dynamo code with the surface flux transport code, we will have a better understanding of the star nearest to us. And in the future, we can hope to extrapolate these fields to and simulate the interplanetary magnetic field in the Solar System.

The purpose of this blog is to post the recent advances in the development of the Surface flux transport code.