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Fair winds!

Dave and Anke
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Saturday, February 18, 2012

Time, Tide and the Rule of Twelfths (or Tenths)




The Rule of Tenths is a decimal approximation of fractions from the Rule of Twelfths... the two are equivalent.
(Drawing not to scale)


Getting around with shoal draft means getting up close and personal with the bottom. Since the tide is busy coming and going, we'll want to keep our eye on it.

The Rules of Tenths and Twelfths are rule-of-thumb approximations. Of what and why are the subject of this post. For now, it's a tool that helps you know ahead of time what the tide is doing at any given moment between high and low. Don't be frightened; it's only arithmetic.
Tides run high to low in about six hours (ebb tide, or the ebb), and back again (flood tide, or the flood) in about the same span. The volume of water flowing in those six hours follows a bell shaped curve. One can divide that curve up into the six hours of the tide. In each hour, a fraction of the whole of this tide's range will come or go. Range is the difference between high and low water:


RANGE = (Height of Water at High Tide) - (Height of Water at Low Tide)

Heights of Water at High and Low Tides can be read in tide tables. They are measurements made relative to Zero Tide Datum; an arbitrary height from which all others are counted. Height of tide may be positive (above ZTD) or negative (below ZTD). Charted depths (heights of bottom), however, are shown in positive units below ZTD. This can lead to ambiguity.

For simplicity, I'm going to speak in terms of heights of water and bottom. I'll reserve depth for the distance between height of water and height of bottom at any given moment, regardless of whether or not it's immersed.



Notice that tidal range, and fractions thereof, will be expressed as some unit of height (feet, fathoms, meters). It is important to keep in mind that these figures measure the height of a change in sea level, and not the height of sea level, itself. Sea level heights are relative to zero tide datum, while range heights are not.

Our graph shows amount or volume of flow. For example the blue column in the fourth hour, for example, represents 3/12, 0.25 or 25% of the total volume of water for one tide, flowing in on the flood, out on the ebb.

At the beginning and end of the tide, no volume is flowing... water is at a stand-still known as slack tide. At high tide, the current is at high slack; at low tide, the current is at low slack.

At the middle of the tide, the greatest volume is flowing. At the middle of a flood tide (incoming), the current is at maximum flood. At the middle of an ebb tide (outgoing), the current is at maximum ebb.

The greater the tidal range, the more volume is flowing in any given hour. The lesser the tidal range, the less volume is flowing in any given hour. Tidal ranges vary according to the Moon (see Neaps Springs Eternal).

Any moment of the tide divides our curve into volume that has come in, and volume that has yet to come in.

*****

Okay... that's the basic picture. Now we get to the numbers. To start with, we pull a little trick to simplify things.

Since water level is rising and falling over an area's entire surface, we may ignore area (and with it volume), and concentrate on changes in height of water, flooding or ebbing, expressed in units of height.

While this trick eases our calculations, it is good to remember that, in this case, height represents volume. A big tidal range generates greater height/volume and therefore currents will be stronger than experienced during low tidal ranges.

A second point to observe is that these hourly changes in height do not represent sea level. Our graph is different than those depicting sea level during a tide cycle. While superficially similar, tidal curves climb or fall for the whole six hours, depending on whether it's flooding or ebbing.

Let's take the end of the 4th Hour of an incoming tide as an example. At that point, 75% of the tide has come in (10% + 15% +25% + 25% for the 1st through 4th hours), with 25% left to go (15% + 10% for the 5th and 6th hours). If the range is 12 feet, then 9 feet will have come in, and 3 feet will be left to go.

Whether I use percentages, tenths or twelfths, the result is the same. The tenths have the advantage over twelfths, in that most tide table heights are given in decimal units (generally feet in US waters). It makes the arithmetic one step easier if we don't have to convert.

*****

We typically have a good bunch of data at our disposal: our draft, heights and times for high and low water from tide tables, depth of water and time of sounding, charted depth (shown below zero tide datum).

Applying the Rules to these data, with different approaches we can answer the following, and more:

What is the height of the bottom?
What is the height of the water at a given time?
When will a certain rock show? How high will it be at a given time?

...These are useful for charting depths. Once water height is established, it's like a vast water level... everything it laps at that moment is also that height. Rock heights are often useful in navigation.

Will we ground out? If so, when?
Will we float? If so when?

...Useful for shoal drafters on a steady basis. Deep drafters benefit if, say, going on the grid. Most cruisers will be letting their tenders go dry on occasion.

How much water will come in before high tide?
How much water will go out before low tide?

...These help with anchor scope and swing calculations. Or deciding how high to drag that tender!

Think of these as puzzles, as story problems, as a challenge. Draw 'em out on paper. Have fun with it. Little more than arithmetic is involved.

With a little thought and practice, you'll be able to do it in your head.


NOTE: There are many sources that help with forming a solid picture of tidal dynamics. I've just touched on the subject here, and there are many further wrinkles to be aware of. Depending on your local, tides may behave somewhat differently than I have described. It's a fascinating subject and worth of a sailor's study! I highly recommend that you learn about educate yourself for the tides of your cruising grounds.

****

Extra Credit Story Problem:

Let's say we're sneaking into a cozy, high water bight where we plan to dry out. We've arrived at the beginning of the 5th hour of the incoming tide (High was 16 feet, low was 10 feet). On arrival, we sound and find that the bottom is 4 feet below the waterline.

Q: What is the height of the bottom?

A: Let's work it out, starting with what we know...

HI = 16ft   (above ZTD)
LOW = 10ft  (above ZTD)
RANGE = HI - LOW = 16ft - 10ft = 6ft   (NOT relative to ZTD)
DEPTH = 4ft   (from sounding, NOT relative to ZTD)

By the Rule of Tenths, we add up fractions for each hour of the tide...

.10 + .15 + .25 + .25 = .75      Of the tide since Low Tide
.10 + .15 =  .25                        Of the tide till High Tide

We apply them to the range to find the change in water height since low and high, respectively...

.75 x RANGE  = 4.5ft     Height of Water since Low Tide, OR
.25 x RANGE  = 1.5ft     Height of Water till High Tide

We adjust our known heights (HI and LOW) by one result or the other, adding to low OR subtracting from high... should be the same result, either way, so you'll only actually be choosing one pair or the other for any given calculation. Let's use 'Height of Water Now' as a shorthand for 'Height of Water at Time of Sounding'...

Height of Water Now = LOW +  Height of Water since Low Tide = 10ft + 4.5ft = 14.5ft, OR
Height of Water Now = HI - Height of Water till High Tide = 16ft - 1.5ft = 14.5ft

Last, we adjust the Height of Water Now for the sounding we took earlier...

Height of Bottom = Height of Water Now minus DEPTH = 14.5ft - 4ft = 10.5ft  (above ZTD)

Voila! We note the Height of Bottom for our hidey hole as 10.5ft on our chart and have a glass of something sippy.

With practice, this will likely be done in your head in about 30 seconds. The important thing is to have a clear picture of what's going on, and go step by step. Don't hesitate to use paper, especially if tired, cold and/or hungry.

4 comments:

  1. Hi Dave,

    I had some fun puzzling over bits of this one. Thought I'd share, in case it's useful to anybody...

    The rule of twelfths is quite familiar to me -- sailing a keelboat in Maine, with an inclination to anchor where there would be just enough water, provided lots of practice. So that was easy, and the conversion to tenths also followed easily. Though, maybe because it's what I learned first, so far I still like twelfths -- I like the way the number of feet of tidal range translates into number of inches for each 12th, which seems to suit my capacities for mental arithmetic.

    I spent more time puzzling over "height of bottom." I understand now, and it's probably obvious to everybody else, but it took me a while to figure out that this was "height of bottom over ZTD (zero tide datum)." A handy calculation, once one has wrapped ones mind around it!

    Another bit that I think made the example a little trickier was the part that the height of low tide in the example was 10 feet. I'm taking this to mean that it's a substantial version of "higher low" in a widely varying higher and lower low tide situation, but it did throw me off for a bit, in getting a good mental image of what I was thinking about, and the meaning of each term in the example. In eastern Long Island sound (Connecticut/New York) the higher high doesn't even reach 5 feet above ZTD, so I had some confusion remembering that 10 feet could be the low end of the tide range above ZTD. Though I can see the reasoning for setting up the example that way, because there is no missing that the sounding doesn't match up with the lowest amount of water to be expected!

    If you were inclined to add another practice problem or two, with answers, some of us would be entertained :-)

    The other thing that caught my attention was the discussion of current, and I wanted to point out for folks who might be unfamiliar, that the timing of slack and maximum tidal current does not necessarily coincide with high and low tide, and halfway in between. I'm quite sure you know all about this... but it seemed worth mentioning for folks who might not be familiar with it. We have so many areas around the northeast coast where those timings are vastly different, as I'm sure you do in Alaska. It's such a great mind bender, to watch the current flowing out of a bay with a narrow entrance, at the same time as watching the waterline on the shore rise! Explained by the water flowing in at deeper levels, while it rushes out on the surface, but still!

    So that's what came to mind -- thanks so much for the discussion. The concept of "height of bottom" has its own shiny new pocket in my collection of anchoring calculations!
    -- Shemaya

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    Replies
    1. Hi Shemaya,

      Excellent points... Sorry about the confusions.

      I'll amend the post to include height ABOVE ZTD. Normally, that would be thought of as 'charted depth', but charts seldom concern themselves with the dryout shoals or even many of the edges. Lots of white space or shoreline apparently drawn from impressions made on the cartographer, rather than actual surveys!

      A complication is that tidal heights above ZTD are given as positive numbers and visa versa. DEPTHS are charted below ZTD, also as positive numbers. Worse, heights of land features are not given relative to ZTD at all, but to another datum, altogether. Sigh. NOAA CHART #1 explains all the fine print. Well worth going over with a fine tooth comb!

      My sample tide numbers aren't unusual in the Pacific NW, and I've not really sailed much elsewhere... didn't even think that the sample numbers might be extreme.

      Right, too about the currents. Things get complex, and simple assumptions are only a starting point! Narrows, too, develop differential based flow that may be well offset from mid-tides on either side.

      I should probably be more explicit that most of these posts are highly generalized, and are only starting points in a lifelong curriculum. It can be a school of hard knocks, too! Nor is what I'm writing necessarily gospel... I try to be accurate, but it all represents the best of my knowledge on the day I write!

      Thanks for helping keep that in sight!

      Dave

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    2. Hi Dave,

      Yeah, I love Chart #1 -- what a discovery that was! And it's those green spaces on the chart, between low water and high water lines, that always seemed so undefined. I'm really delighted with this tool you've shared for calculating those heights from soundings. It looks so simple AFTER it's been explained! Thank you for taking the time to write it out and post it.

      As for currents and their timing, what an interesting bunch of material they present. And then there are rivers -- here, where the Connecticut River nears the ocean, the flood current runs upstream for about four hours each cycle, and the ebb current runs downstream for about eight hours each cycle -- even though the tide height rises and falls every six hours like one would expect. When the river is really running after major rain or snowmelt, the flood current (on the surface) never happens -- but the ebb-direction flow gets milder for a few hours. I knew all about this theory for ages, but didn't really wrap my mind around it until kayaking downstream for a couple of hours, with an expectation of coming back up on the flood current. Ha! Ever since THAT experience, the whole thing has been clearer :-)

      Cheers,
      Shemaya

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    3. Hi Shemaya,

      That vertical stratification of current happens a lot up here in the long Straits and Passages, as well as tidal estuaries.

      I've been chafing to try a technique I read as used by the Hubbards (drifting the Mississippi in their barge riverboat... book is SHANTYBOAT by Harlan Hubbard).

      He called it a MUDSAIL... it's a square sail fixed at the bow and let down into the water(!), weighted along its foot with pipe or similar, control lines led aft. Use it to run in the current, even into the eye of a foul wind!

      Should be able to lower it into deeper strata of fair current when available to overrun a foul surface current.

      BRILLIANT!!!

      Dave

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