Year 12 Maths

Part of my job as a teacher of meteorology is to go to NZ universities looking for future meteorologists. It breaks my heart when, sometimes I meet a person with a genuine passion for the weather who would love to work for us as a meteorologist, but just can’t cope with the required maths. Unfortunately for them, professional meteorologists need to have some university maths under their belts. And this requirement isn’t just a local thing – it also comes from the World Meteorological Organisation of which we are a member nation.

Why is maths important in meteorology? Let me give a small example to illustrate, related to the thread of my previous blog posts, wind…

On weather maps we see isobars which are the lines going around each anticyclone (area of high air pressure “H“) and depression (area of low air pressure “L“). Have you ever noticed that the isobars seem to spread out more in the anticyclones than in the depressions?

How isobar spacing varies around anticyclones and depressions
How isobar spacing varies around anticyclones and depressions

To check out this effect, take a look at the real-life example below (covering a large part of the southwest Pacific – I have undisplayed the fronts), or have a look at a few current weather maps to see for yourself.

Mean Sea Level analysis, midday 4 July 2006
Mean Sea Level analysis, midday 4 July 2006

This spreading out of isobars in anticyclones is a fairly general result, and the reason comes down to the solution of a quadratic equation. If you did maths at high school you might remember these. I learnt about them in the fifth and sixth forms (now called Year 11 and Year 12). These equations look something like this:  ax² + bx + c = 0, where a, b and c are given to you and you then have to work out what “x” is. You can find out lots more about them on the internet, e.g. Wikipedia, or just ask your kids!

When our trainee meteorologists study weather, they learn all about wind, isobars and air pressure, as well as the effect of the Earth’s rotation. It turns out that there is a strong connection between these factors, and it is described by a quadratic equation (I won’t write the equation here, but you can ask me for it through the comments section below if you’re interested).

The solution, x, of this particular quadratic equation holds the key to the answer, because x represents the wind speed. In this case x behaves differently depending on whether you are in an anticyclone or a depression. For anticyclones there’s a limit to how close the isobars can be and therefore a limit to the wind speed, depending on how close to the centre of the anticyclone you get (there is a small latitude effect too). However,  for depressions there’s no limit to the closeness of the isobars, provided the atmospheric forces are big enough to drive the air pressure into that configuration.

I think it’s an amazing and powerful result, and it all comes down to school-level maths! This really is just the tip of the iceberg as far as maths and meteorology are concerned – you could spend a lifetime studying solely the meteorological applications of mathematics.



The quadratic equation has been requested as in the Feb 2011 comment below. The equation is:

which can be solved for V:



When I was biking to work in Auckland  last Friday I noticed prolific amounts of pine pollen in the puddles around Westhaven.

This was probably blown here from the pine trees of Riverhead forest during that northwesterly gale we had the previous night.  That was a vigorous cold front — producing wind gusts to 120 km/h, and a period of heavy rain which produced flash flooding in Greymouth,  and slips in Wellington enough to stop the train to Masterton.

The graphs below show the wind from Mount Kaukau NZKKW and the peak of the Rimutuka Road NZRIX  as well as Manukau Heads on Auckland’s West Coast NZMKW. ff  =average speed,  fm= gust speeds.


The top graph shows the average winds ff (no more than gale force) and the gusts fm. Timestamps are in UTC, so 23 0000 is noon Thursday and 24 0000 is noon Friday local time.


Weather forecasts usually just mention the average winds, but also mention the gusts when they become significant.   Our Severe Weather Warnings for strong wind give the gust speeds expected.  From these people can judge the likelihood of damage by using our wind poster.  In this graph wind speed is in knots.  Multiply by 1.85 to get km/h — the top gusts plotted here are 65 knots at Kaukau and 66 knots at Manukau heads (around 120 km/h).  Yes, it was windier (just) over the hills of Auckland than it was in Wellington.

The bottom graph shows wind direction, with wind blowing from feathered to pointed end, each barb worth ten knots, north up and south down.   I’ve included it so we can see the wind swing as the cold front moved across the North Island, most noticeable at Mount Kaukau which shows the southerly gale going through Cook Strait on Friday. Rimutuka Hill road is farther north and, during Friday morning, slightly to one side of this Cook Strait southerly.

Usually I notice “pollen in the puddle” around the start of August, so this year it seems to be around a week earlier than normal.  While it is another sign that spring is coming, it can not be taken as a sign that winter is over.

As an old saying puts it, describing the weather at this time of year:  “As the days get longer, the cold gets stronger”.



Picture credit

Will Richie McCaw and the ABs have the weather on their side or in their faces for this weekend’s big game?

On Saturday night at Eden Park in Auckland it’s the opening game in the 2009 Tri Nations rugby competition, and the Bledisloe cup is up for grabs as well, and New Zealand’s weather is turning nasty –again.

Auckland’s last three weekends have been wet and this coming weekend looks set to continue this cycle.   Spare a thought for Wellingtonians – they are likely to get their eleventh wet weekend in a row (mind you, they only had 0.2 mm during Queens Birthday weekend).

MetService has issued Severe Weather Warnings for heavy rain in the eastern Bay of Plenty and Gisborne hills and ranges and are keeping a watch on possible heavy rain and some snow for the northern Hawke’s Bay ranges.  This is associated with a low-pressure system which is expected to deepen as it moves past the northeast of New Zealand during Friday night and Saturday.  Its rain band should move over northern areas tonight, dampening down Eden Park after a few dry days.


MetService Weather map at 6am Friday

That rain band is likely to dump around 80 to 120 mm over the eastern North Island ranges during Saturday—but in Auckland , as the low moves away, a strong , gusty southwesterly wind is likely to prevail, blowing away the rain and replacing it with showers, ensuring a slippery field.  As Eden Park regulars know, these showers come and go, and by the time the game starts on Saturday evening they are likely to be spaced further and further apart.

As far as the weather is concerned it is very likely to be a “game of two halves”:  one with a shower and squally wind to boot, and t’other without.

The wise team will be checking the MetService radar an hour before kick-off to decide, if they win the toss, to run east towards Mount Eden in the first half or delay this privilege until after half time.

As for the spectators, dress warmly as we are forecasting the air temperature during the game to drop to around 9C and the wind chill during that likely shower to feel as if its around 3C.

That’s cold, but not as cold as the weekend forecast for the South Island.  A southwest blast straight from the Southern Ocean is likely to bring gusts of around 110 km/hr to Southland and Otago on Saturday night with snow lowering to near sea level and some thunderstorms.

Challenging weather for a challenging game. Bring it on!

Blooming daffodils and wayward winds.


MetService’s Auckland office is in Westhaven, and often I can be seen at lunchtime wandering around Victoria Park.  Much to my delight yesterday I came across some daffodils blooming outside one of Auckland’s oldest pubs (photo). 

and daisies
and daisies

And near the flyover (which is currently in the news) some white lawn daisies are already showing themselves.

These are wonderful signs that warmer days are coming.  But not quite yet! Late July to early August is, on average, the depths of winter in New Zealand.   It is the period with the coldest days- the season for curling.

Last night, Basil, one of my sailing buddies, commented on how he had a wonderful time in the squadron race last Saturday.  “What happened to that harbour gale warning?” he asked me.  I know that many fellow Aucklanders may also be interested in the answer, so here goes:
The above graph shows the wind speed average as measured in knots at three of the Coastguard Nowcasting stations around the Hauraki Gulf (maintained by MetService).  The time stamps are in UTC, so 11 0000 translates to noon Saturday, and 12 0000 is noon Sunday.

Channel Island (yellow) increased to storm force (over 47 knots) by noon on Saturday and Tiritiri (blue) to gale force, as correctly predicted in the MetService recreational marine forecast .

Bean Rock at the harbour entrance only increased to 25 knots (strong) and for much of the time was less than that.

Why so?  Friction is part of the explanation—wind can blow smoothly over the sea but when it blows over land the extra friction can knock around a third off the average speed.  MetService refers to wind speed in its land forecasts in km/hrand for marine areas in knots – this way people can tell that they have the right forecast for land or sea.

Another factor is air density. The day started with a buffer zone of cold dense air hugging the ground over Auckland.  The incoming gale was coming from the southeast, and blew over the Coromandels and then over the top of this buffer zone, unable to dislodge it.  The gale blew over the city, and for a brief time it dipped down to touch the Skytower, but it never dropped to sea level.  In this case the gale couldn’t dislodge the antecedent cold air.   This is rare, and it is more reasonable to assume the gale would get to sea-level rather than not, hence our forecasts.

MetService checks all its Auckland marine forecasts and we have a monthly average accuracy of 94%.  OK, this one didn’t work out and we will mark it as wrong.  In this case the gale didn’t get to sea level.   Please don’t assume that this happens often.

For example: check the following graphs:
These show what happened at Whangarei on the same weekend. The coastal wind, as measured at Tutukaka (in yellow), rose to storm force on  Saturday afternoon, peaked  near midnight, and eased below storm force late Sunday morning, all as forecast, and with similar timing and intensity to what was happening about the outer Hauraki Gulf.

In this case there was no mountain range upstream from Whangarei to help disrupt the wind, and a gale was able to touch down into the city.  The winds at the airport (in red) were knocked back a third by friction (as per normal) but still reached gale force.  That was consistent with the damage reports around the area, including fallen trees and flying branches taking out powerlines.

I’ve also included the barometer and rain readings from Whangarei.  Note the close but not-quite-mirror link between pressure and wind speed.  This is typical for a low like this,  so if you have a barometer you can use it, in cases like this, as a rough wind speed forecaster.  Also note the peak in the rain occurred as the low was still approaching and eased off before the barometer “bottomed out”.  This is also fairly typical, but each low-pressure system is slightly different in this regard.

I find it fascinating that there is so much pattern in our weather, and I find it frustrating when, sometimes, chaotically, one of them behaves slightly differently, as in this case producing wayward winds for Auckland.

The Great Northwesterly Storm of August 1975

If you were in New Zealand in the mid ’70s you may remember a particularly strong wind-storm that devastated many parts of the eastern South Island. It struck on 1 August 1975, doing a huge amount of damage to pine trees in the Eyrewell and Balmoral forests in particular.

To give you an idea of the power of this storm, some of the peak recorded winds and gusts were:

  mean wind (including gusts and lulls) strongest gust
Christchurch Airport 126 km/h 172 km/h
Timaru Airport 130 km/h 165 km/h
Eyrewell Forest 119 km/h 170 km/h

Compare these winds with the kinds of winds you hear about in most severe weather warnings these days. MetService issues severe weather warnings for widespread damaging winds when the mean speed is expected to reach 90 km/h or gusts to exceed 110 km/h.

I recall this storm as a schoolboy in Christchurch, and the local radio station broadcasting that school was closed for the day. Looking out our front window I remember seeing a boy trying to cycle to school, barely able to stay on his bike. At lunchtime there was a short period of rain (from the cold front drawn on the weather map below), followed by a pleasant sunny afternoon with a gentle wind.

Mean Sea Level analysis, 6am 1 August 1975
Mean Sea Level analysis, 6am 1 August 1975

Because I was interested in meteorology even then, I thought about this storm, wondering why it was that Canterbury and Otago suffered most of the damage rather than Westland. Surely, I thought, the wind that had blown over the mountains must have come from upstream, so the wind must have been equally strong on the western side of the Southern Alps? But this was not so; the wind on the West Coast was only half as strong.

The fallacy in my argument was my implicit assumption that our weather and the forces that drive it are only two-dimensional (2-D). Perhaps we inadvertently reinforce a perception that our weather is 2-D by publishing lots of weather maps that are valid only at the Earth’s surface (e.g., see Weather Maps). Rarely do we show what’s happening higher up through the atmosphere or, more precisely, the troposphere – the part of the atmosphere that contains our weather.

It was the 3-D character of this storm that forced the damaging winds in Canterbury and Otago. Let me explain why. Firstly, we live in the mid-latitudes so most of our weather, especially high up in the troposphere, comes from between northwest and southwest (this is not always the case but it’s generally true; for example, take a look back at the satellite loop in my previous blog post).

Secondly, we also know (see Mt Kaukau blog post) that the wind is usually stronger higher up. This effect was marked on 1 August 1975, when a northwesterly wind became very strong over the top of the Southern Alps.

And finally, there was an area of low air pressure at ground level on the leeward (eastern) side of the Alps.

Strong wind over the mountain tops descends on the leeward side.
Strong wind over the mountain tops descends on the leeward side.

These three factors together led to damage on the ground because the very strong wind aloft descended into the low-pressure area. It was this 3-D descending flow, and the turbulence that accompanied it, that did the damage in the Great Northwesterly Storm of August 1975.

In my next post I will continue this thread by again looking at wind but from another perspective.