Longitudes, latitudes and the real length between two points on earth

Earth is a sphere. It’s not a perfect sphere but to make the life of most of the engineers and scientists easy we usually make the assumption that it’s a perfect sphere.  At a glance it might seems that calculating the length between two points on a sphere is a very easy task, but ladies and gentlemen, it’s not.  It took me several hours to figure it out.

First we have the great circle formula.

In the figure GCA is the great circle angle. It is the angle that is created by the two axis that can be drawn from the two points to the center of the earth.

The great circle Formula to find out the great circle angle

Cos(great circle angle) = Sin(latA) Sin (latB) + Cos (latA) Cos (latB) Cos (longA-longB)

The derivation of which is given in the  site below

http://faculty.chicagobooth.edu/alwyn.young/research/Papers/MinimumSphericalDistances.pdf

The great circle formula is getting simplified for the points in the same longitude and the same latitude

For the same longitudes it becomes

Cos(great circle angle) = Sin(latA) Sin (latB)

For the same latitudes it becomes

Cos(great circle angle) = Sin^2(latA)  + Cos^2 (latA)  Cos (longA-longB)

Then we can measure the  distance between any two points using the formula below, where Rearth is the radius of earth

L=Rearth X  great circle angle in radians

Avoiding polar singularity in ocean modelling.

 In general singularity is  a point at which a given mathematical function is not defined. When numerically modelling oceans we come across with the two poles that can be considered as singularity points. 

There are several methods to deal with the singularity points. However in my work we are only concerned about the Arctic pole. We deal with it by rotating the North pole in to the equator. It should be mentioned that this rotation affects the Coriolis force.  I will discuss later how to rectify this affect.

To rotate the coordinates -90 degrees about the y axis we use Euler rotation Matrix. Given the rotation angle is tita the matrix can be represented as below.

Rotation Matrix

Congelation ice formation direction

I took a bit of time to understand this phenomena. Once I understood it, it became really interesting to me since I was working on boundary layer computations in my masters.  Before describing the phenomena let me describe some keywords that would make it easier to understand it.

Congelation ice: This is the ice that grows by freezing in to an existing ice bottom that has already formed in a calm sea.

c-axis: It's the reference axis perpendicular to the plane of a movement of rock or minerals

boundary layer: the fluid layer adjacent to a solid.

corrugated:  a series of parallel ridges and furrows- The free dictionary

Corrugated ice

According to the observations of Langhorne, there seems to be a correlation between the ice formarion direction and the direction of the current. Let's consider the following two figures.

(a) current direction is perpendicular to the grooves parallel to C axis

(b) current direction is perpendicular to the c-axis and parallel to grooves

In the figure (a) current direction is perpendicular to the grooves. This creates mixing in the boundary layer  between the ice and the water. The boundary layer is slightly turbulent. We can also assume that the salinity of this boundary layer is quite high since freezing emits the brine in to the water. The mixing in the boundary layer changes the salinity in figure (a) that makes it more favorable in ice growth. In figure (b) since the groves are aligned with the current direction, we could assume that the boundary layer is laminar and therefore no mixing takes place. It isn't favorable for ice growing. Therefore we  can conclude that ice growing has a tendency to align the c axis parallel to that of the current direction.

Reference: 

Ice in the ocean, Peter Wadhams

Challenges in Data Assimilation

In my research what I'm trying to do is to prove that data assimilation would be an effective way to gain better results in ocean numerical modelling specifically in the sea ice. However today I read an article about the errors arise from data assimilation. It was an eye opener.

There are three types of errors


1. Temporal error:

This is the kind of error that arise from mismatch of time and space.  For an example the value of temperature that is being measured might be a point data while the data used by the computation model depends upon the time step, usually in my case it's one day. This also applies for the position of data obtained. Matching the grid points and actual position of data obtaining points is quite challenging.

2. Instrumental error:

This is the error rising from the error in instruments. The observational data that we use in assimilation might be wrong and the error can get accumulated with cycles.

3. Assimilation Residuals:

The above errors might be easy to avoid with proper handling of data measurement but residuals are inherent to the concept of data assimilation. Data assimilation tends to nudge the results towards observation results. This might lead to avoid the extreme weather predictions.

Other than these errors, it must also be considered that obtaining observational data is extremely difficult in oceanography.

Ocean Stratification

Stratification refers to the layered nature of the ocean and the atmosphere. Ocean is consisting of several layers. This is a picture of the several layers lies in the ocean.

The mixed layer between the atmosphere and the ocean is called Marine boundary (mixed) layer. The layer from the surface until 100m deep is called the Mixed layer. Most action takes place in the mixed region since this is the layer that gets heated up by the sun.
from 100m 1000m is the isothermal layer. The layer below the isothermal layer is called Thermocline. The temperature decreases in this region with depth. There's very little vertical motion in this region

Ref
http://stream2.cma.gov.cn/pub/comet/MarineMeteorologyOceans/ocean_models/comet/oceans/ocean_models/navmenu.htm

Sigma coordinates.

Sigma coordinates are widely used in oceanography due to the differences in the elevation of the seabed.  The elevation can vary from several meters deep near the coast and several kilo-meters in the deep basins. The regular z coordinates are mostly compatible only with uniform basins.

z is the vertical length from water surface to any point. H is the depth of the total water column. η is the sea surface height that is affected by the tidal force. Sigma varies from 0 to -1 where -1 is at the  bottom earth surface where z=-H and 0 is at the top of water surface z=η.

From Wikipedia

 In sigma coordinates system the number of vertical levels in the water column is the same everywhere in the domain though the depth of water column is different from place to place. The Navier Stokes equation will then be represented in sigma coordinates.

Zonal and meridional

They are used to describe the directions on earth

Zonal refers to the direction from east to west

Meridional refers to the directions from North to South of earth

Zonal temperature or meridional wind flows are some commonly used terms

Reference:

http://en.wikipedia.org/wiki/Zonal_and_meridional

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Arctic dipole anomaly

There is a pressure difference between the two regions of the Arctic. High pressure on the North American region and low pressure on the Eurasian region give rice to this Arctic dipole. Arctic dipole leads to drive more Southern winds in to the arctic ocean which affects the melting of sea ice. This said to be the reason behind the recorded sea ice minima during the past few years.

Reference
http://en.wikipedia.org/wiki/Arctic_dipole_anomaly

Ice albedo feedback

You can read about Ice albedo at my previous post. In simple terms ice albedo refers to how much light is being reflected by the ice. Ice albedo feedback however refers to a positive feedback climate process.

A positive feedback climate process is a  process in which the outcome is being amplified by a feedback. In this scenario when ice albedo decreases ice absorb large amount of radiation and it starts to melt. when it melts ice albedo is further reduced and it starts a vicious cycle that increase the melting of sea ice

Reference:
http://en.wikipedia.org/wiki/Ice-albedo_feedback

Ice Albedo

The first new word I found is Ice albedo. This comes from the two words Ice and albedo.

Albedo is the reflectivity. In simple terms it is the measure of how white something is. It is also a measure of how much light does something absorbs. The albedo scale ranges from 0 to 1, where 0 being perfect absorption (pitch black) and 1 being perfect reflection (As white as white can be).

Talking about ice albedo it ranges from 0.5 to 0.7 which is quite high reflectivity. On the other hand ocean albedo is around 0.06 in other words ocean absorbs most of the visible light. Snow has the albedo of 0.9 therefore it acts as a protecting layer over the sea ice.

When snow melts it creates melting ponds that has the albedo of 0.2 to 0.4 in that case albedo drops to 0.75 when the melting pond deepens the albedo can drop to 0.15

You can do further reading at
http://nsidc.org/cryosphere/seaice/processes/albedo.html

Where to start?

As I've mentioned before I've got a title. Yey!!

"Incorporating data assimilation in numerically predicting sea ice to assist navigation in the Arctic Ocean"

I also need to take classes as well. This semester I'm just taking three classes but until add drop period I'll be taking nearly 5 classes. I hope I'll be able to figure-out the classes I'm taking soon enough so that I can focus on my real research.

Right now I'm taking Marine Environment modeling class, also I take propulsion engines, Coastal environment studies, Urban engineering. That's a nice mix. The first one was suppose to be in English but the lecturer decided to do it in Japanese, Therefore I might want to drop the class later. My favorite out of all the classes is propulsion engines. It's about space propulsion engines. It's damn interesting.

 Getting back to my title, the key words are data assimilation, numerically predicting sea ice, Arctic ocean. My knowledge about them is really low. But one time I read in an article, that getting a PhD and being an expert in an area are two things. Therefore I'll not make the same mistake as  in Msc by going in to the details of my minor. I'd rather try to find answers to the following questions.

1. What is data assimilation?
2. What data assimilation methods are most suitable?
3. Which parameters should be used in data assimilation?
4.Where in Arctic should I focus first?
5. What is POM?
6. What is ICEPOM?
7. How are they compiled?
8. How does one create a mesh, assign initial conditions and boundary conditons and run a simulation in the cluster using ICE-POM?

These are to be answered during the first few months. First of all I'm gonna read the manual of ICE-POM to get started.

At the very beginning of the journey

Many people start PhD with absolutely no clue. Some people continue their Masters to the PhD, which makes their journey quite easy. I on the other hand have a title.

"Incorporating data assimilation in numerically predicting sea ice to assist navigation in the Arctic Ocean"

Having a title is a good thing, because during the first year of PhD many students are killing themselves to come-up with a suitable title. However during my entrance exam one professor was showing his doubts towards the novelty of my research. Data assimilation is not a new concept, however applying it in to modeling sea ice might be the novelty of my work.

During my masters I worked on Computational Fluid dynamics. My main application was airplanes. Therefore I believe I have some skills in CFD in aerodynamics. However for my PhD my application changed to ocean modeling, I find it very interesting since the results are more visible and I can get a feel of what I'm doing.

Writing this I believe would help me to keep a track on what I'm doing. I'm planning to update this at least once a week. I was hoping to do the same for masters but honestly couldn't find enough time.

Well I'm quite optimistic about the coming 3 years. I would very much like to publish a journal paper by the end of first year and much more in the coming three years. I've also not forgotten the remarks of my good friends who took the same path and were trying to stop me from ruining the last bit of my twenties among clusters and professors.

Well, I took the decision to do a PhD by myself because I believe I'm also suffering from closure syndrome. It would kill me to not to finish the last bit of the educational journey. Also in my opinion one should either stop at Bsc or at PhD. So here I'm after 22years of education, is starting the feared journey. If anyone is jobless enough to read this, don't forget to wish me luck.