Your first sentence is correct. It's an example for Conditional Zero.

You use conditional zero when talking about something which is true in general. You can tell from the distance between the rings whether there was a lot of rain or not. This is always true.

The pattern for Conditional Zero is:

If + Present Simple, Present Simple

You can replace if with when or if not with unless in conditional zero. For example:

Unless there is much rain in a year, the rings in a tree are close together.

Here's a link to an overview on Wikipedia about all conditionals in the English language.

For the sake of completeness, "will + inf" as you use it in the second sentence is used in Conditional 1. As you can read on Wikipedia, Conditional 1 is about "consequences of a possible future event".

1. A better rephrasing.

• Narrower tree rings indicate seasons of lower rainfall.
  (FYI: some trees have multiple seasons--and multiple rings--in one year.)

  Here's some culled from the Internet:

• Drought decreases growth, producing a narrower ring. Source here.

• This layer, or ring as seen in cross section, can be wide, recording a wet season, or narrow, recording a dry growing season. Source here.

• Google Search: tree rings narrow wide

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2. Both are Zero Conditional: statements of general fact.

Both sentences mean the exact same thing: A general truth statement that low rain in a year produces rings that are "close together" (like shown in the picture). In this case, there is no significant semantic difference between using "are" and "will be". Either sentence can be substituted for each other, and it will convey the exact same thing.

On the other hand, there are better ways to word this concept, as the other Answerers and I have pointed out. The sentences contain conflicting elements of grammar, semantics, multiple senses of time (both semantic and grammatical), and domain specific knowledge. All of these elements create forces that pull in different directions. If anyone wants to know the details, I can discuss offline.

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3. Structure, Semantics, and Grammaticality.

This section reveals some relationships between structure, semantics, and grammaticality. Eliminating "in a year" makes the sentences more difficult (but not impossible) for the reader to contextualize. The reason "why" is demonstrated in the rest of this section.

The original sentence contains a lot of complicated semantics of time--like "close tree rings" which requires two rings on either side of a narrow ring. By eliminating all mention of trees and rings and measurements, we can see that the sentence itself is in the form of a general truth statement:

• If there is not much ABC-stuff in a year, the PDQ-observables in an XYZ-System [are | will be] close together.

Let's choose another real topic that's not so complicated:

• If there's not much food in a year, the birds will be unhealthy.

Since "a year" is non-specific, the sentence is talking about any year. Also, in sentences like this, we understand from it's own context as a truth statement that the domain of truth excludes exceptions such as some birds may be dead instead of unhealthy. And it's obviously not speaking to the case of a severe famine in which all birds die. Also note, a truth statement doesn't have to be "true":

• If there's not much food in a year, the birds will be happy and healthy.

That's still a sentence that is grammatically valid. It just happens to be false.

Now consider dropping the phrase, "in a year". You're right that in a year helps to make the time-sense match either verb. Without in a year, both of your sentences (moreso I think for the "are" form) loses some degree of grammaticality/acceptability.

• If there's not much food, the birds will become unhealthy. (Grammatical.)
• ? If there's not much food in a year, the birds will be unhealthy.
• ?? If there's not much food , the birds will be unhealthy.
• ??? If there's not much food, the birds are unhealthy.

In this notation, the question marks indicate a subjective sense of degree-of-grammaticality, called acceptability. (Whether my judgement is "perfectly right" is not the issue here; I just want to focus on that last sentence, which I explain further next.)

The last sentence isn't perfect, but it's still comprehensible A lack of food results in unhealthy birds over a period of time, and the removal of "in a year" eliminates this supporting notion. It's easier to understand the last sentence when "resulting over a period of time" is understood implicitly:

• A bushman who speaks English is leading an ecologist through the jungle where the food supply had been running short that season. He spots a recently deceased bird and picks it up. Softly parting the feathers and revealing the sores on bird's skin to the ecologist, the bushman says, "If there's not much food, the birds are unhealthy." The bushman then drops the bird and, using his walking stick, continues with a sure yet gentle march into the deep wood. The ecologist frantically studies the bird with rubber gloves and snaps some pictures, knowing that he must hurry or be left behind.

Let's simplify the sentence even further:

• ?If it rains, the grass is healthy. (The grass doesn't suddenly become healthy if it rains, but we can "excuse" that and understand the truth of the sentence in the context that it is true.)

• When it rains, the grass is healthy. (The use of "when" introduces a vague sense of time. Still, if we think of "when" meaning "the day it rains" then it doesn't make sense. But "when" makes it easier for us to interpret a longer time period, which better fits the context in which the statement is true.)

• ??If the sun shines bright, the ground is warm. (We know that it takes time for the ground to become warm. The sun can shine bright early in the morning without the ground being warm. To accept this sentence as true, we have to dismiss a lot of exceptions.)

• ?When the sun shines bright, the ground is warm. (Again, the use of "when" is more ambiguous and can be interpreted as a longer time frame. It makes it easier to dismiss some of the exceptions.)
• When the sun shines bright, the ground becomes warm. (This is better because we don't have to struggle against the implication of an instantaneous change from cold to warm. We can easily accept this as "true" because we can easily imagine the scenario in which it is true: a normal sunny day during the summer a few hours after sunrise.)
• When the sun shines bright, the ground becomes warmer. (This is the easiest sentence because we don't have to attribute any particular temperature as "warm". Even if it's a frosty cold day and the ground changes from 0° to 5°, it's not "warm" but it is "warmer". However, even in this case, we have to think of this in terms of when it is true. If a freezing cold front sweeps through, the ground could become colder despite a brightly shining sun.)

4. What if we drop "in a year"?

Finally, we can get to the root of the issue. The form of these sentences is not a cause-and-effect truth statement. They are Truth Statements that TimeProcess2 co-occurs-with and is-dependent-on TimeProcess1:

• When TimeProcess1 occurs, TimeProcess2 co-occurs.
• When TimeProcess1 occurs, the TimeProcess2 end-state is achieved.
• If TimeProcess1 occurs, the TimeProcess2 end-state is achieved.

As long as we can think about "the truth" from the deictic time orientation that occurs exactly when the entire process has completed to TimeProcess2 end-state (which, deictically, is thought of as "now"), all three sentences are perfectly logical!

Imagine now, standing on a warm road during a sunny day. You can truthfully say all three of the following sentences:

• TimeProcess1 = Sun Shines Brightly; TimeProcess2 = Road becomes warm.
• When the sun shines brightly, the road becomes warm. (Non-deictic truth statement.)
• When the sun shines brightly, the road is warm. (As I notice it "now".)
• If the sun shines brightly, the road is warm. (A proposition is made deictically.)^1.

One can say this same Zero Conditional truth statement from any deictic perspective:

• When the sun shone brightly, the road became warm. (From the perspective of the end-of-time, this is a Zero Conditional truth statement!)
• Whenever the sun shone brightly, the road became warm.
• When the sun did shine brightly, the road did become warm.
• When the sun will be shining brightly, the road will be becoming warm. (From the perspective of the beginning of time, this is a Zero Conditional truth statement!)

Now finally, note what happens if TimeProcess1 ends before TimeProcess2 ends.

• Statement 1: *If you turn on the stove, the water is boiling.

  The asterisk indicates this sentence is not grammatical (grammaticality)--or extremely low acceptability. This is due to the unresolveable semantics of time. Similarly: *Let's go to the circus yesterday.

  Turning on the stove ends well before the water boils.

• Statement 2: *When you turn on the stove, the water is boiling.

  Not grammatical.

  The use of "when" does not allow us to ambiguously extend the time frame to the point of the water boiling.

These last two sentences clearly breaks the form! When the semantics of "When" does not allow us to ambiguously extend the time frame, we can no longer supply any scenario that allows the coinciding of the two processes.

This demonstrates that the reason we can make sense of your example sentences is due to the ability to imagine a "when" scenario that allows the processes to coincide. So the phrase "in a year" supplies some semantics that **helps one to imagine the coinciding processes, and conversely, eliminating that phrase makes it more difficult.

5. The fifth dimension.

We can actually create a perspective in which both Statement 1 and Statement 2 (above) are logical and understandable! From the hyper-deictic 5th dimension perspective in which we can see all time and space existing "all at once". We can see the "flow of time" is imaginary, and we can see both the turning-on of the stove and the boiling of the water. Now the sentence means something a little different:

• When you turn on the stove, the water is boiling.

What that means is we are looking at the time-space slice that is true because we can see it; we can see the entire process all at once. It's true not because of any cause-and-effect. It's true because it's a directly observable fact.

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^{1. If the sun shines brightly, the road is warm. (A proposition is made deictically.) This is perhaps one of the most fascinating parts of this Answer. We might just dismiss this statement as "slightly odd". But I think it's better than that. This is not the normal way we think about making propositions. But there's actually nothing inherently "wrong" with thinking of propositions this way! It's a different kind of time-view paradigm in which the "now" the only "time" in which we can make any propositional statement, and our "so-called normal" way of making logical statements is an imaginary abstraction away from reality. I'm going to call this deictic propositional logic.}

Note: if you are in a hurry, you can just read the conclusion toward the end.

Both sentences are grammatical. And the grammar of each can be used to state something that is always true. I am going to call this kind of statement a STIAT.

STIAT is formed from the first letters of: Something That Is Always True. I will also not use the word "c _ _ d _ _ i o n a l" in this answer. And I will try to avoid using the word "if."

I am going to look at a variety of ways that a STIAT can be expressed. Hopefully, by the time I've done this you will see that you can use either sentence to express a STIAT (something that is always true). Or I hope at least that you will be mildy amused at my feeble attempt to do so. And remember that STIAT refers to something that is always true, grammatically speaking.

If there is not much rain in a year, the rings in a tree are/will be close together

The sentence you ask about in your question has two clauses:

1 a subordinate clause (If there is not much rain in a year,) and 2 an independent clause (the rings in a tree are/will be close together).

We will call the the subordinate clause, P; and the independent clause, Q. When a STIAT is expressed with this type of sentence, you get this: P always means Q.

P always means Q can be expressed in many ways. We certainly do not need the words if and then. In many stores you can find a sign

YOU BREAK IT
YOU BUY IT

The native shopper speaker knows that this means

P: You break it
Q: You buy it

'You break it' always means 'you buy it'.

I.e., You break it : You buy it. YBIYBI Or: If you break it, you buy it.

It is a STIAT or something that is always true. At least that is what the sign says. And that is what the grammar of the sign means.

This is so true that there are other signs that go far as to say:

You break it
You bought it

'You break it' always means 'you bought it'. Or in a sentence:

If you break it, you bought it.

Note that although this is a fully grammatical sentence, it is somewhat of a colloquialism. It is the case, however, that the past tense can be used in the Q clause of a conditio- this kind of sentence.

This conceives of the {act of buying} something you break as a past event the moment you break it. What is indicated here is not some weird time sequence, but the inevibility (or STIAT-ness) of YBIYBI ('You break it' always means 'you buy it'.) So much so, that the moment you break something, you have already bought it, as far as the store is concerned.

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We can also express a STIAT using the modal will in the Q clause. One can do this in any number of quaint proverbs or idioms, such as these:

Boys will be boys. (more info)
When the cat's away, the mouse will play. (info)
A drowning man will clutch at a straw. (info)
Faith will move mountains. (info)

and two well known laws.

1 The Field of Dreams maxim

If you build it, he will come. (Watch the 30 second clip on YouTube.)

and

2 a straightforward version of Murphy's Law.

If anything can go wrong, it will.

All these sentences are STIATs. If the P clause is fully expressed you get:

Given the chance to be Boys, [they] will be boys.
When the cat's away, the mouse will play.
Provided that one is A drowning man, he will clutch at a straw.
Assuming that you have Faith, it will move mountains.
If you build it, he will come.
If anything can go wrong, it will.

Again, notice the use of will in the Q clause.

These proverbs state "timeless truths" or things that never change or things that are always true, or, in the singular, something that is always true (STIAT).

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One can state a STIAT in the following way also:

Provided that the temperature of water gets to 0 degrees, it freezes.
Assuming you drop oil in water, the oil floats.
As long as it rains, things get wet.
When you hear a noise upstairs, that's Dad falling out of bed.
Assuming you leave early enough, you get to the airport on time.

Notice the use of the simple present in the Q clause.

You can also state any of these STIATS using a different verb. Here's how: You take your mouse button and copy the five sentences, then you paste them below. Then you change the verb in the Q clause from the simple present to the modal "will" plus bare infinitive. Here, I've done it for you:

Provided that the temperature of water gets to 0 degrees, it will freeze.
Assuming you drop oil in water, the oil will float.
As long as it rains, things will get wet.
When you hear a noise upstairs, that'll be Dad falling out of bed.
Assuming you leave early enough, you'll get to the airport on time.

IF you need to see these in the traditional if, please do:

If the temperature of water gets to 0 degrees, it freezes.
If you drop oil in water, the oil floats.
If it rains, things get wet.
If the hear a noise upstairs, that's Dad falling out of bed.
If you leave early enough, you get to the airport on time.

and

If the temperature of water gets to 0 degrees, it will freeze.
If you drop oil in water, the oil will float.
If it rains, things will get wet.
If you hear a noise upstairs, that'll be Dad falling out of bed.
If you leave early enough, you'll get to the airport on time.

CONCLUSION:

Both sets of above statements, ie, whether you use the simple present or will + bare infinitive, you are/will be expressing STIAT (something that is always true).

In Meaning and the English Verb Geoffrey N Leech says: "Oil floats on water and Oil will float on water are more or less equivalent statements."

Therefore, whether you write your sentence as

If there is not much rain in a year, the rings in a tree are close together

OR

If there is not much rain in a year, the rings in a tree will be close together

does not matter.

They both express P always means Q. They both talk about something that is always true. They just do so in different ways. But we have already seen that we can talk about P always means Q in several ways.

Removing in a year
There is not any change in the meaning of your sentence if you remove in a year other than the obvious: you are no longer referring to tree-ring growth of a given year, you are now just talking about tree-ring growth in general.

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Sources include:

Meaning and the English Verb, By Geoffrey N. Leech (2004). Most of the quaint proverbs taken from here.

The Teacher's Grammar of English: A Course Book and Reference Guide, with answers, Ron Cowan (2008). Cambridge U Press.

C-ndit-onal Sentences at this Berkeley website. Especially germane is:

Content-based conditionals are understood by relating the content of the two clauses to each other. A typical way in which content conditionals can be understood is for the "P" clause to identify a situation which causes or automatically results in the state of affairs signalled by the "Q" clause. This is the case for

If you drop it, it will break.
If you say that again, I'll slap you.
If it rains, we'll cancel the picnic.
[emphasis mine]

Thank you for your time, and I hope that this answer has been somewhat helpful.
{The above phatic line not to be deleted until 1/3/16.}