{"id":23280,"date":"2013-09-26T09:21:22","date_gmt":"2013-09-26T15:21:22","guid":{"rendered":"http:\/\/rankexploits.com\/musings\/?p=23280"},"modified":"2013-09-26T09:21:54","modified_gmt":"2013-09-26T15:21:54","slug":"23280","status":"publish","type":"post","link":"https:\/\/rankexploits.com\/musings\/2013\/23280\/","title":{"rendered":"Oscillating heating: Toy Problem for Ocean Heat Content."},"content":{"rendered":"

This post is a “gedunkan” to illustrate something that happens in simple systems. It’s prompted by a discussion in comments in the discussion about the conversion of Ocean Heat Content to equivanlent Temperature Change<\/a>. This is not intended as explaining the complexities of what happens in the ocean, but merely to show a feature that makes it difficult to unambigously interpret what is happening in the depths of a body based on data about the total heat content in the lower and mid layers of a body heated from above.<\/p>\n

The Gedanken<\/b>
\nIn this Gedanken or “toy” problem, we will example a very deep solid body that is heated from above. This body could be made from solid block of any very good heat conductor (e.g copper, aluminum, gold) which we insulate at the sides. We will assume the sides are perfectly<\/I> insulated. The very bottom of the block will be dunked in an stirred ice water bath keeping it the bottom surface of the aluminum at T=0C. At time t=0 we assume the temperature of the aluminum is T=0. The conceptual diagram is show below:
\n\"VeryDeepPlate\"<\/a><\/p>\n

In our mind we will partition the upper layer illustrated and call it “atmosphere”. (For the purpose of the gedanken, ignore the fact that the earth’s atmosphere is not aluminum. We can discuss in what ways this problem is similar or different from “the earth’s climate system” in comments.) The next layer down is the “top ocean”, the next down is the “top 2000 meters” and the lowest bit is “the lower ocean”. We will not explore what it means for the bottom of the ‘ocean’ to be kept at 0C.<\/p>\n

Now we start the actual gedaken: at the top of the aluminum block we place an element that can either heat or cool the top of the block. This element provides a controlled heat flux of Q= 0.1 cosine (πt) where t is time. That is illustrated with the red arrows. Note that the heat flux is positive for 0<t< 1\/2 then it goes negative. This is shown below:
\nHeatAtTop<\/a><\/p>\n

\"HeatAtTop\"<\/a><\/p>\n

Below, I’ve illustrated how temperature varies with depth at three different times, two at times < 1\/2; the third a bit afterwards.
\n\"TemperatureWithDepth\"<\/a><\/p>\n

The earliest time illustrated is shown in blue, as nearly everyone expects, the heating at the top causes the temperature at the top of the block to rise. Because the bar is conductive, heat propagates downwards. The red trace shows the progression in temperature over time. Because heat flux at the surface remains positive in the time frame from the blue trace to the red one, the temperature at the surface continues to rise, ahd heat continues to propagate downward.<\/p>\n

Finally, the green trace represents a time just after 1\/2, when heat flux at the top surface turns negative<\/i>. What happens during this time is a bit complicated. At the very top surface, heat is sucked out at the top. So this tends to cool the top of the block. At the same time<\/i> recall that at t=1\/2, the temperature at the top<\/I> of the block is higher<\/I> than the temperature lower down. So, at that point, conduction acts to suck heat away<\/I> from the top and down to the bottom. So, the “top” layer loses heat both<\/I> because the heat flux at the top turned negative and by conduction to lower layers.<\/p>\n

When creating the green trace, I chose a point in time when the top three points had lost sufficient heat such that the maximum temperature is now between the third and fourth triangles on the left hand side of green trace. For those who think ‘this doesn’t happen’: Yes it does. Go into a desert before sunset. The top of the sand will be hot, dig a bit: the sand lower down will be cooler. Go later: the top will have cooled, but if you dig a bit, the sand lower down will be warmer than the top. This happens.<\/p>\n

Returning to the figure: At the point in time represented by the green trace, conduction will act to transfer heat from points closer to the maximum point toward those further away. So: to to the right of the 4th point from the left, conduction will cause heat to flow deeper<\/I> into the block while for points to the left of the peak it will cause heat to flow upward<\/I> toward the surface. We could show more and more traces. But for now, I won’t. Instead, I’ll discuss the relevance to the discussion of interpreting what it might mean<\/p>\n

    \n
  1. A top layer cools<\/li>\n
  2. A mid layer warms<\/li>\n
  3. The lower layer warms (possibly even faster.)<\/li>\n<\/ol>\n

    Note that in my conceptual figure of the block, I divided the region into “atmosphere”, “top ocean”, “mid ocean”. Suppose we were to mentally place the division bewteen atmosphere between between the three green triangles on the left and the forth to the left:
    \n    \"TemperatureWithDepth\"<\/a><\/p>\n

    In that case, it’s fairly easy for someone integrating by eye to determine that the mean temperature (or heat content) of the top layer would cool slightly, while those of all lower lower layers warm during the time period from the “red” trace to the “green” trace. It happens that if I continue this exercise, depending on the choice of where I place the “mid” and “lower” layer partitions, I can get all sorts<\/I> of behaviors. <\/p>\n

    I can have the <\/p>\n

    (top layer cool, mid layer warms , lower layer warms less quickly than midlayer)
    \n(top layer cools, mid layer warms slowly, lower layer warms more quickly than midlayer)
    \n(top layer cools, mid layer cools, lower layer warms) <\/p>\n

    and so on. And I’ve only just started the oscillating heat application, and (because EXCEL bogs down) I haven’t didn’t make by block deep. Let’s suppose I continue to apply oscillating heat. Look at the temperature variation with depth at time 2 which is the first full cycle of heat appliation:
    \n    \"LaterOn\"<\/a><\/p>\n

    Given this illustration of what happens in a very very simple<\/I> physical system, I think you might want to contemplate what one might conclude if the only<\/i> thing one knows about a “haitus” in surface heating is the “ocean” to heats when the atmosphere displays a “hiatus” from previous heating. I would suggest that unless someone fills in more blanks, it’s difficult to conclude much at all. Because if the heat is applied from the top and stalls, this is precisely what happens with no need of introducing any fancy theory about “enhanced mixing” at lower layers, or “the top didn’t really stall”, or “heat is hiding” or whatever. It’s just what happens in a simple system. <\/p>\n

    Some will wonder: Could I make this problem more complicated and still show the main result? Sure. But I can’t simultaneously make it simple<\/i> and sufficiently complicated to match the dynamics of the ‘earth ocean’. If I add complications, that can cause people to think the behaviors has somethign to do with the complications when, in fact, the behavior is a sort of leading order very simple behavior. So, for discussion, I think this is best. The purpose is merely to let people see this and reflect. If you have complications that interest you, I might be able to explore some of the simpler ones. (Though, I have to admit, I did this in EXCEL which is behaving like a lard-ass cranky program. )<\/p>\n

    Still, discussion is welcome. <\/p>\n","protected":false},"excerpt":{"rendered":"

    This post is a “gedunkan” to illustrate something that happens in simple systems. It’s prompted by a discussion in comments in the discussion about the conversion of Ocean Heat Content to equivanlent Temperature Change. This is not intended as explaining the complexities of what happens in the ocean, but merely to show a feature that … Continue reading Oscillating heating: Toy Problem for Ocean Heat Content.<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[314],"tags":[],"class_list":["post-23280","post","type-post","status-publish","format-standard","hentry","category-toy-physics"],"_links":{"self":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/23280","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/comments?post=23280"}],"version-history":[{"count":0,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/23280\/revisions"}],"wp:attachment":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/media?parent=23280"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/categories?post=23280"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/tags?post=23280"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}