On the Hovmoeller plot

In this post I will explain one of the standard graphical methods that we in meteorology, oceanography and climate science use, the Hovmoeller plot. We use this when we want to visualize phenomena that move in time and space, but we don’t want to have to look at (or make) a movie. We want to see the motion in a single, two-dimensional picture. We often use the Hovmoeller plot to visualize the MJO, and I will focus on a particular Hovmoeller plot that is used in MJO forecasting and research.

For example, here is a Hovmoeller plot that was produced by the DYNAMO forecast team as part of this week’s forecast. (Sorry it's a little blurry, our team of graphics experts here at Madden-Julian Conversation is on vacation this week.) This specific plot was produced by Matt Wheeler at the Centre for Australian Weather and Climate Research in Melbourne, Australia.

This plot shows, in colors, outgoing longwave radiation anomalies, averaged from 7.5S-7.5N latitude, as a function of longitude and time. Time is on the y axis, and runs downward, while longitude is on the x-axis. Projections on the MJO and other coherent tropical modes are superimposed in the contours. Let’s break that down one piece at a time.

Outgoing longwave radiation (OLR) is the energy flux (amount of energy per area per time) in infrared radiation coming up from the earth into space. Because it’s infrared, it is basically heat radiated by the earth and the atmosphere (i.e., it is not reflected sunlight, which is mostly visible light rather than infrared). How much radiation is emitted from something depends on what that something is made of – some things are good emitters and some things are bad emitters – and on its temperature. The warmer something is, the more radiation it emits. Radiation is emitted from the earth’s surface itself, land or sea; from gases in the atmosphere, especially water vapor; and from clouds.

If there are no clouds, the atmosphere is partially transparent. Most of the gases that make up the atmosphere are not great emitters, with the most important exception being water vapor, which is a relatively good emitter. So most of the radiation comes either from the surface, or from lower layers of the atmosphere where there most of the water vapor is. The atmosphere gets colder the higher you go, so the surface and lower layers are relatively warm compared to the upper atmosphere. Thus the OLR is high in clear skies, because the radiation is coming from warm places. On the other hand if there are thick, high clouds – say, the kind that go with heavy tropical rain - those are really good emitters, and also good absorbers. So they block the radiation that would otherwise come up from below, and the radiation the satellite sees is what is emitted from the tops of the clouds. The OLR will be low in this case, because the tops of the clouds are at high altitude, where it is very cold, so they don’t emit as much radiation as they would if they were lower down where it is warmer.

So to summarize, low OLR = high clouds, and probably rain; high OLR = clear skies. Low OLR is blue in the picture, high is red.

What’s shown in the picture is not the total OLR, but the OLR anomaly. If you were to just plot the total OLR, a lot of what you’d see would be the “climatology”, meaning “what is normally there”. The climatology is is computed by taking the average from many years of data for that place at the same time of year. Some places tend to have a lot of clouds and rain (e.g., Indonesia) and they would be blue, while others tend to be dry (e.g., Arabian peninsula) and they would be red. But we don’t need data now to tell us what normally happens at this time of year, we are interested in what is happening now. We want to know how that is different from the climatology. So we take the OLR at a given time and place, and subtract from it the climatology, to get the anomaly. Note that “anomaly” here doesn’t mean “something really unusual”, as it does in normal English usage; it just means “the difference between present conditions and the average of what has happened at this place at this time of year in the past”.

Ok. So there is an OLR anomaly defined every day at every point on the map – mathematically, it is a function of three variables: latitude, longitude, and time. That is too much information to visualize in a two-dimensional picture, so we average in latitude. We are interested in the deep tropics, right near the equator, so we average the OLR anomaly between (in this case) 7.5 degrees south and 7.5 degrees north.

The result is a near-equatorial OLR anomaly that is a function of longitude and time. Those are the two axes on the plot, longitude and time, and the color represents the OLR anomaly value. Blue means rainier than usual, red means drier than usual. Now, by tracking diagonal bands of color, we can see weather patterns moving. WARNING: MATHEMATICS ALERT. Call longitude x and time t. Now, remember your algebra: a line on the graph would the form t=ax+b, where b is the intercept (which in this case tells us the time at which the line hits the Greenwich meridian, at zero longitude) and a is the slope. Such a line describes a trajectory, something moving in longitude with time at a constant speed. The slope of the line is inversely related to the speed. In other words, the shallower the slope, the faster the line moves, as it crosses more longitudes in a shorter time. Because time runs down in this plot, a line sloping down to the right (a<0) means motion to the east, while a positive slope (a>0), means something moving to the west.

In the lower left corner of the plot, you see a ragged blue blob sloping down to the right; that’s the active phase of our recent MJO event. Let's point it out:

It was at 40E, in the western Indian ocean, around mid-october, and reached 120E, around Indonesia, just recently. This blob slants downward such that it takes 10 days or so to cross the Indian ocean, which corresponds (if you do the math) to a speed of a few meters per second. Above it, you see an orange-red blob with the same slant, which is the suppressed phase of the MJO that preceded the active phase.

The solid horizontal line running across the whole plot in early November represents the date on which the plot was made. Anything above that line is an observation of what happened, while anything below it is a statistical forecast. What is a statistical forecast? Basically it is a forecast based on the assumption that weather patterns will continue evolving as they normally do, based on past experience. In the case of the MJO, this means that a region of low OLR (rainy) that starts in the western Indian ocean and moves eastward at the typical speed (again, a few meters per second) will continue to go eastward at that speed into the western Pacific. Here, look at the blue contours that lie on top of the blue shaded region. Those contours result from a mathematical smoothing of the OLR anomaly, in an attempt to extract just what is the MJO – it’s the “envelope”, or large-scale pattern one might see if one let one’s eyes go unfocused, as opposed to the misshapen rougher features that indicate random weather variability within the envelope. The statistical forecast is essentially produced - as you can see by eye - by letting the contours of the envelope stick out below the date on which the forecast was made, with the same slope that they had before that date.

What are the other contours on the plot, some of which slant down to the left rather than to the right, and which have different slopes indicating different propagation speeds? These are other coherent tropical wave modes, known as Kelvin waves, Rossby waves, and others. We'll write about these in a later post.