Figuring out tides

As seen in the previous post, the tides can be complicated due to the resonant properties of embayments, the Coriolis effect and the like.

You might wonder how people figured tides before NOAA or tide charts, or before Laplace.    Old nautical atlases had just about every piece of information a navigator could want.   Not only were there charts, but also what the appearance of a stretch of coastline looked like, and long descriptions of how to carry out navigation using the instruments available.  In these nautical atlases were tide tables - not the kind that you might get at a bait-and-tackle shop, but rather it had lists of the relative timing of high tides to the position of the moon in the sky.   Nominally, we might think that the high tide occurs when the moon is at the highest point in the sky, but this isn't always the case.   With rotary tide systems, in some harbors, the high tide will be synchronous with the rising moon, or the setting moon.  

These tide tables would list the ports in Europe for which they were good for, and then list the number of days after a full moon, and then give information for the time of the high tide relative to the position of the moon in the sky.   This turns out to be a relatively robust way of describing and predicting the tides, although it lacks the precision of a full-blown harmonic analysis that one finds in NOAA charts.

Let's say you show up on an unknown coast and have to figure out what's going on with the tides?  The first question is the phase of the tide, and whether it's coming in or going out.   Also, the next question is "where is the moon"?

Usually you can tell what part of the tide cycle you're on by looking at what's uncovered.   This is location dependent, as there are different life-forms present in sandy shores versus rocky shores.   The part of the shore that covers and uncovers with the tides is called, appropriately, the intertidal zone.   Excepting extraordinary tides, sea grass does not grow in the intertidal zone, so that marks the upper limit of the tide.   Typically the high tide line is marked by detritus of some kind - dead, dried out seaweed, old party balloons, driftwood, and the like.   Due to the neap/spring tide cycle, you'll often see multiple lines of detritus.

Below this, there are often shells from dead mollusks, like periwinkle shell and clam shells.   Below this line, if there are rocks present, one often finds barnacles attached to rocks.   The lowest zone is when you typically begin to see seaweed emerge as bright green.

If you get to a beach, you can first try to guess what part of the cycle you're on.   Then, wait 10 minutes or so to see whether the tide is coming in or going out.   Depending on what's exposed, you can take an educated guess how far away you are from low or high tide, depending on how far away the water line is from sea grass, or how many periwinkle shells are exposed etc.  During the time of high tide or low tide, there's about an hour where no motion will be visible and, again, one can take an educated guess for the timing.   The correlation of this time with the position of the moon can allow you to track the tide, providing you know something about the time with respect to the moon (e.g. knowing where it is in the sky at high tide and advancing high tides by 12 hours 24 minutes each cycle).

Likewise rocky shorelines, like one finds in Maine can also give information about the tidal cycle.   Again, detritus forms the uppermost line at high tide.   Below that, one finds first slippery rocks and rocks with barnacles on them.   A wide swath of the intertidal zone has a kind of seaweed called rockweed on it.   This has distinctive air bladders on it, which cause it to float up.   Often, if you peel away rockweed from tide pools, mussels can be found.  At the extreme low end of the intertidal zone, one begins to see kelp, which can be quite slippery to walk on.  As before, a bit of sleuthing with the moon can give decent results in predicting tides.

Most of us don't have the time to scout out shorelines and figure out tides, so getting advance information is usually the norm.   In the U.S. one of the best sources is NOAA's website:

http://tidesandcurrents.noaa.gov

If you go to the upper left hand side of the page, there's a drop-down window called "Products".   Under "Tides/Water levels".  If you go to "NOAA Tide Predictions",   there are two options, one is a google map interface, where you can select a location where the tide is predicted by clicking on the location.   The other option is a menu of locations, categorized by region and then state.   If you then click on a state, you'll get a listing of stations that have tide predictions, looking something like this:

Listing of tide stations on NOAA's website. 

Harmonic stations have all the harmonic constants available, like the amplitude and phase of M2, the semi-diurnal lunar forcing.   The subordinate stations use an interpolation of harmonic stations to make an 'educated guess' for the tides.    If you pick one station, you'll get a graph that looks like this:

Graph of tides from NOAA's website for Cutler Naval Base, Machias Bay, Maine. 

In the above graph, you can see the tide height relative to the MLLW datum (see previous post for definition).    The great thing about the NOAA tide chart is that you can get the height of the tide for any time.    On the other hand, people typically only remember the time of the high and low tide and don't print out and carry around the full graph (although I do some of the time).    

Another case is when you get a table of just the high and low tides in a table.   Below is a snippet from the Boston.com website with high and low tide times for Hull, on the south side of the Boston Harbor complex.  

Typical tide table listing just high and low tide times and heights relative to MLLW. 

If you look at the time of the high and low tides in Hull, you'll see that the timing isn't precisely 12 hours, 24 minutes between different highs.   For the morning (10:34 AM) high to the evening high (11:13) on Jan 1st 2014 is 12 hours 39 minutes, but the evening to morning time difference is 12 hours 14 minutes.   Likewise, the heights of the highs are different - 11.7 ft and 10.3 ft above the MLLW.  This is due to different harmonic components coming into play.   These aren't big departures from the 12 hour 24 minute nominal timing between highs, but it gives you a rough idea of the variations from a precise semi-diurnal period.

Rule of twelves

There is a way of figuring tides in between the highs and the lows called the rule of twelves.   This works reasonably well for semi-diurnal tides and diurnal tides, and can be adapted, although with a bit more care, to mixed semi-diurnal tides.   The height of tide over time resembles a curve called a sinuisoid - it's a smooth curve where the name comes from the trigonometric function.  

The 'rule of twelves' is shown below.   There are roughly 6 hours between high tide and low tide for a semi-diurnal tide.   The change in the height of tide happens in ratios of 1:2:3:3:2:1 over each of the 6 hours after high tide.   The sum of 1+2+3+3+2+1=12 is easy to remember.   At the end of the first hour after high tide, the height drops 1/12th of the height.   By the end of the second hour, it drops 2/12 from value at the start of the second. Then it drops by 3/12 by the end of the 3rd hour, and then 3/12 again by the end of the 4th hour.   The rate begins to diminish and drops 2/12 by the end of the 5th hour, and finally 1/12 by the end of the 6th hour.   

Illustration of the rule of 12ths.

If one is in an area where there are tidal currents, the strongest currents will be during the mid-point between high and low tide - roughly 3 +/- 1 hours after the high or low tide.   The rule of 12ths can also be used to figure out roughly the strength of the current throughout the tidal cycle.   

We can try the rule of 12ths on the Hull tide table.   Note that high is 11.7 ft at 10:34 AM, and that low is -2 feet (relative to the MLLW datum).   This is a range of 13.7 feet - a bit more than 12, which is our round number - so one "unit" would be something more like 1.2 feet. This would mean that we'd get the following tides, roughly:

10:34 - 11.7 feet

11:35 - 10.5 feet

12:40 - 8   feet

 1:45 -  4   feet

 2:50 -  1.5 feet

 3:55 -  -1.5 feet

 5:00 -  - 2 feet

This is kind of by eye-balling it - not using a calculator and realizing that there was more like an hour and five minutes for each "tidal hour" to make the low tide come out to 5:00 PM for low tide.   Note that I didn't try to do any better than a half foot in my estimation. Below is the actual chart from NOAA.   I tried to line up the times (dotted lines) as best I could with the graph and you can see that I did pretty well with my estimates from the rule of 12ths - not perfect, but within about half a foot. 

Comparing my estimates from rule of 12ths to NOAA data.  

Currents

In addition to figuring out the heights of tides, which can be helpful in making landings and ease of launching, tide cycles are important as it helps one predict currents.    Tidal currents can be quite powerful.    The strongest known tidal current is the Saltstraumen near Bodø Norway.   Current speeds up to 25 knots have been clocked there.    While this is pretty unusual, tidal currents of 2-6 knots can be quite common in different parts of the world and one has to plan crossings and paddles accordingly.    Another effect is the combined effect of wind and current.   If wind is blowing against the current, it can created sizable waves.   These waves have something of a different character from plain old windblown waves, and can be somewhat trickier to negotiate - more erratic and with steeper faces.   Many times it's easiest to surf the waves if one is going against them.   Again, crossings can be aided by knowing when on the tidal cycle the currents will be weakest. 

Charts will sometimes have currents indicated on them with a little arrow.   Below is a piece of a chart indicating the waters to the southeast of Grand Manaan Island, New Brunswick.   This is at the entrance to the Bay of Fundy.   The chart has an arrow in the lower middle of the figure with the arrow indicating the direction of the flood tide.   Some text in the chart indicates that the tidal currents can be 4 to 6 knots, with warnings about a very heavy current on the ebb. 

Portion of nautical chart to the southeast of Grand Manaan Island, New Brunswick. 

Charts don't always carry current information.  One can rely to a very limited extent on trying to figure out what's going on knowing the tidal range and using charts, but this can often be misleading. Again, NOAA website on currents and tides is a great source for the United States.  From the main page of tides and currents, you can again pull down the "products" drop-down menu and go to tidal current tables.  

Below you can see a screen-show of the tidal current table for Casco Passage at the entrance to Bluehill Bay, just to the southwest of Mount Desert Island.  At the top, you can see the name and the month that the table is valid for (June 2014).   The "Day" just means the date of June (June 1st, 2nd...).    In the heading, it gives you the direction of the flood, in this case 88 degrees true, almost due east, and the ebb direction, prefaced by a minus sign, and is 284 degrees true.   

Tidal current table for Casco Passage at the east end of Blue Hill Bay. 

If you look at the table, it starts out with the time of slack water listed in a 24 hour clock-time, referenced to the local time (LST/LDT).   The time of maximum current is listed along with its speed. The + sign means it's on flood and the - sign means it's on ebb.   In the case of Casco passage, there currents aren't huge - of order 0.7 knots maximum, not enough to strike fear, but enough to make one compensate for the current to make a crossing in the fog.   

Generally NOAA and other sources of tidal currents do not offer as much granularity as tide heights and this can be something of a challenge for sea kayakers.   First, since only the times of the maximum currents are given, one has to interpolate to the time of a crossing in order to estimate the currents at any moment.   

There are multiple ways of figuring out the currents between max flood or ebb and slack water.   Like the rule of 12ths, these are all simply ways of making approximations to the sign curve at different hours into the tide cycle.   When the height of the tide is changing the most rapidly, this is when the current runs the strongest. 

 Perhaps a more important factor is the fact that the tidal currents are not reported with as much geographic granularity as tide heights, which creates challenges in figuring out local currents.  

Generally speaking one can inspect a number of the tide tables and associated currents to get a rough idea of the currents associated with different locations and then interpolate both geographically and temporally to take a good guess as to the tides one might encounter.    

Let's take a look at a navigation exercise to get concrete.   This is probably more complicated than you have to make things, but it gives an idea as to how you might guess a max current when a current reporting location isn't available. 

Below is a chart of the entrance to Machias Bay.   Say you're camping on Ram Island and with to make the crossing to Cross Island.  It's completely foggy, so you have to rely on a ferry crossing.   First, you make your way up to the northeast corner of Stone Island and then want to paddle directly to the westernmost tip of Cross Island (Northwest Point).   This is a crossing of about two nautical miles.   How do you figure out the tidal currents and your heading in order to head directly across?   Let's say you are allowed the luxury of trip planning in advance and merely have to anticipate conditions.   The date is June 16th 2014 and the time you leave Stone Island is 12:00 (noon). 

Chart for navigation exercise at entrance to Machias Bay. 

So, the first task is to figure out what the current speed and direction is at noon on June 16th at the entrance to Machias Bay.   We can first look for the timing of the tides.  Fortunately, there is tide data for Stone Island, which is shown below for June 16-17.   You can see that at noon, you're about 2 hours before high tide.    Presumably this is a flood tide, but what's the current speed and direction? 

Tides at Stone Island for June 16-17 2014

The nearest stations where currents are reported are Moosabec Reach, Jonesport, Friars Road, Eastport, and 7.6 miles SSE off of Pond Point.   Pond Point is on the southeast corner of Great Waas Island, so it is considerably out to sea.   A clip of the tidal currents for June at this station is shown below.   The flood is 15 degrees true and ebb is 215 degrees true.    The values for max flood is 10:57, which is about midway through the tide cycle at Stone Island.   The max current for this time is 0.7 knots on the flood.   

Tide current tables for the station 7.6 miles SSE of Pond Point, Great Waas Island. 

Now this is when things get a bit tricky.   The general trend for a flood current to be to the NE and ebb to be to the SW is common in this region of downeast Maine.   The other two closest stations Moosabec Reach E and Friar Road, Eastport.  The tidal current tables say that Moosabec Reach E.  floods at 110 true and ebbs at 258 true.    The max current on the 16th is 1.4 knots on the flood on the 16th.     On June 16th in Eastport, the flood current is 3.2 knots, with a direction of 210 degrees true.    What do we make of this? 

Well, first, Moosabec Reach is a constrained channel, so one would expect that the shape and orientation of the channel has a lot to do with the current direction.   Since it's constricted, one would also expect a higher current.    Friar Road in Eastport is likewise a highly constrained channel and the channel dictates the direction of flow.   In general, as one approaches the Bay of Fundy, more to the NE, the current picks up, so it's no surprise that the current in Eastport is much higher than either Moosabec Reach or the Pond Point station.   

As for the entrance to Machias Bay, one would expect that the currents would be larger than Point Point because of the Bay draining.   Since currents run to the NE on the flood, one might also expect that as they turn the corner around the region of Stone Island into Machias Bay that they'd pick up there.   So, the combination of being closer to the Machias Bay, more to the NE of Pond Point, that you'd get a higher current between Stone Island and Cross Island than the Pond Point predictions. 

So, what to do?   Generally with higher tides, currents get stronger.   For example, on June 16th in Eastport, the difference between high and low tides are 23 feet and the max current is 3.2 knots on the flood.   On the other hand, on June 6th, the tidal range is 15 ft in Eastport and the flood tide is 2.2 knots.   Being the nerd that I am, I tried to take tide range on June 6th and 16th at Casco Passage, Moosabec Reach, and Friar Road, and correlate the flood speed on both days with the tidal range.  This is shown in the figure below.   

Table of tidal range in feet for June 6th and 16th for Casco Passage, Moosabec Reach, and Friar Roads in Eastport.  

If I plot all six points, I get a somewhat linear relationship between the three, as can be seen in the figure below. 

Flood speed in knots as a function of tide height in Casco Passage, Moosabec Reach, and Friar roads in Eastport. 

Now, this is overly nerdly on my part, but I can actually fit a straight line to this data, and when I look at the tide range at Stone Island, the linear fit predicts about 2 knots off of Stone Island.   Now, Moosabec Reach, Friar Roads, and Casco Passage are more restricted than Stone Island, so I might back off on that assessment to get a max current on the flood of about 1.5 knots, but not much less.   Also, looking at the shape of the drainage out of Machias Bay, I'd reckon that the current would flow roughly parallel to the long axes of Stone Island and the Libby Islands on the chart.   This is a flood direction of 30 degrees true or 48 degrees magnetic, allowing for a magnetic variation of 18 degrees locally.   So, 1.5 knots max current on flood at 30 degrees true is my prediction.

  The animation below is done by Josko Catipovic.    It shows the area between Jonesport and the entrance to Machias Bay.   If you read the vertical scale, it looks like I might have underestimated by a bit the current just off of Stone Island.   I'm guessing from the animation that the maximum current is about 1.7 knots, so 1.5 isn't such a bad guess.  

Tide animation courtesy of Josko Catipovic. 

So, having established a good guess at the direction of max flood and its direction, we need to figure out what the current is at noon.   From the tide graph, we can see that low tide is at 8 AM.  High tide is at 2 PM.   Noon is four hours into the high-low tide cycle.   One would expect that maximum flow to occur at three hours into the tide cycle or at 11 AM, so the question becomes - what to expect for the current one hour after max flood.    Below I show a figure that has the same rule-of-twelves for the height of the tide, and then below that the corresponding current over the different hours.   For the mathematically inclined, the tide in my representation is a cosine and the current is a sine.   All this really means is that the time when the height of the tide is changing most rapidly corresponds to the time when the current runs the strongest.   

Roughly speaking for each hour of the tide cycle, this curve can be represented by the sequence 0-50%-90%-100%-90%-50%-0 or in terms of fractions 0, 0.5, 0.9, 1.0, 0.9, 0.5, and 0 for each hour in the cycle.   This is shown below. 

Tide height and current as a function of time into the tidal cycle.  

Going back to the chart, the desired course heading is 68 degrees.   I'll assume that one paddles at a speed of 3 knots.   Now that we know the maximum current (1.5 knots), we need to figure out what the current is at the time of crossing (noon).   High tide is at 2 PM and low is at 8 AM.   So the crossing is four hours into the tidal cycle.  Using the above rule of thumb, being an hour past the max flood, you'd expect 0.8*1.5 kts = 1.35 kts at noon.   Now, whether we call it 1.35 or 1.4 kts is pretty immaterial, since there will be much larger variations from place-to-place in the crossing and my estimate of the max flood current is probably only good to 10% at best.    But, I'll work with 1.35 kts anyway, carrying more significant figures than I probably should.   

With this information, we can figure out the ferry angle.   The figure below is based on the construction I outlined in this post. 

Construction to figure heading to achieve desired course bearing of 68 degrees true. 

From the figure you can see that the heading needed to make the desired course bearing is 84 degrees (true), and the resulting speed is 4 knots.   The crossing distance is 2 nautical miles, so the crossing should take about 30 minutes and when I factor in the current, my course bearing should be 68 degrees.

What about uncertainties in current?   I'd say I can't do any better than 1/4 knot in predicting the current, once you get done with all the local effects, and may actually do worse than that.   As in the dead reckoning exercise , one has to plan on uncertainties in heading.   I'll take the 1/4 knot and look at the high and low sides: 1.6 kts on the high side and 1.1 on the low, but keep the same heading of 84 degrees.  

In the figure below you can see the range of angles associated with the uncertainties I assigned to the currents.

Range of course bearings to Cross if the current was mis-estimated by +/- 1/4 knot. 

Looking at the range of angles I might encounter, you can see that there's a chance I'd overshoot the northern end of Cross Island if the current was higher than my estimate.   Based on that, I'd adjust my heading by adding 10 degrees so that the most likely landfall on Cross is in the middle of the peninsula.    So, this means a heading of 94 degrees true would give me (roughly) a course heading of 78 degrees true - guaranteeing a decent landfall on Cross.   Once you reach land on Cross, hand-rail to the north, following the coast. 

Just to close the loop on magnetic versus true, I worked the above exercise using true compass headings.   To convert my 94 degree heading to magnetic, I use the CADET rule (see this post) and realize that I add 18 degrees (local variation) to 94 to get my magnetic heading of 112 degrees.   So, when departing the NW point of Stone Island, I follow a magnetic heading of 112 degrees, and the combined effect of that heading, my paddling speed, the current set and drift should place me squarely on the western end of Cross Island.   

It's not necessary to do all the above work in planning a crossing, but I was extra-careful in the process to illustrate the point that there is some uncertainty in the current speed, and there are even other factors that I'm going to mention that will show even more uncertainties.   The idea that the current picks up as you go farther downeast in Maine is the principle point I wanted to make.   I don't expect people to do linear fits, but the progression of max flood speeds from 0.8 knots in Casco Passage on that date, to 1.4 in Moosabec Reach to 3 near Eastport suggests that my estimate of a max flood of 1.5 knots coming into Machias Bay on max flood was a decent guess.   

As I mentioned in the dead reckoning exercise, there are many factors that come into play in a crossing.   In this case, I wanted the most direct passage, modulo a deliberate compass offset to hit the peninsula.   

Here are some other factors that come into play:

1.) The currents may depend on the distance from shore.   In some cases, the drag of the bottom will slow down currents near a shoreline.   In some cases, the current might actually be stronger as they turn the corner into an embayment.   This happens to be the case with Stone Island, as the Islands in that cluster and the peninsula jutting out at the western end of Machias Bay has the effect of intensifying the current speed. 

2.) In an area with a lot of islands, the current can be highly local and predicting conditions depends on being able to read what the water will be doing with different bathymetry.  The figure below depicts the current flow in the area to the south and east of the entrance to Machias Bay on the flood as best I can capture it, based on my experience in the area. 

Local current conditions in the area to the south and west of the entrance of Machias Bay. 

In the above marked-up chart, you can see that the general flow well out in the Gulf of Maine is from the west-southwest, as it floods toward the Bay of Fundy, but local effects come into play.   A major one is the current turning toward the north to flood into Machias Bay.   On a smaller scale, there is a shallow channel just to the north of Ram Island (indicated on the chart).   Here the channel itself directs the flow more to the east.   Since it's shallow, there's less cross-sectional area for the water to flow and it can pick up quite a bit in that channel.   

Another effect you can see in the islands just to the east of Rogue Island (the C shaped island to the west on the chart).   The flood is directed between these islands and runs to the north, fanning out in "rooster tails" on the far side of the openings between islands.    Right now, it's almost impossible to model the bathymetry and water flow to such detail, so some ability to "read" the general flow can reduce the surprises you encounter.   

Finally, here's a real-life experience in this area to illustrate the need for flexibility in planning and having some abilities beyond hoped-for conditions.   You can refer to the charts above to follow along.    I was on a paddling trip out of Jonesport.   We camped at the abandoned Coast Guard station on the north shore of Cross Island.   In the morning, the plan was to wait for the tide to drop down from flood and launch as the flood was waning, in the hopes of crossing to Ram via the Libby Islands more or less at slack tide.  

We launched and paddled in a light current out of the channel between Cross and the Cutler Peninsula, just to the north.   We turned to the southwest, crossing to the Libby Islands, but as we approached the Libby Islands, the wind really picked up - I'm guessing maybe 15-20 knots with gusts up to 25.    The wind was out of the southwest.   We decided to take a break from the wind by finding a landing spot on the eastern side of Libby.   This was a welcome break.  

We paddled back around the north the Libbys after the break, but the extra time we took resting had put us into the ebb part of the tide cycle, and now we had a current ebbing to the southwest and flowing against a wind from the southwest, so the wind was against the current.   This created a fair amount of waves.   We didn't really work on a ferry angle, as it was clear, although I did take note of the compass headings that would give us a direct passage.    Paddling was a little sloppy in the chop, but we finally made it to Ram Island in one piece.   

This experience just illustrates that one has to be ready for conditions that may be more severe than what you planned.