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How are sentences like 日本がピンチだ (Japan is in a pinch) and 明日は雨だ (tomorrow it
will rain) analyzed?

Translating them trivially, as

> 日本がピンチだ Japan is a pinch  
>  明日は雨だ Tomorrow is rain

doesn't make sense. So something must be going on. Is it

  1. 日本 and 明日are topics, not subjects

This would only make sense if が is interpreted as a topic marker, which would
be an expansion from its usual interpretation: a subject marker. "This
wouldn't be a problem for the second example" you might say, but e.g.
subordinate clause 明日が雨だったら is valid, and has the same problem.

  2. ピンチ and 雨 are の-adjectives

> ピンチの国 → 日本がピンチだ  
>  雨の日 → 明日は雨だ

  3. Loose/alternate interpretation of copula

You might argue that the Japanese copula doesn't equate things to the same
degree as the English one, but merely associates them. This might seem the
most straightforward explanation, but looking at [the wikipedia article on
copula](http://en.wikipedia.org/wiki/Copula_\(linguistics\)), such
interpretations are not mentioned.

  4. Something else?
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grammar particle-が particle-は copula
Analyzing sentences like 日本がピンチだ and 明日は雨だ
Your translations are wrong.

  * The Japanese word `ピンチ` in this usage is not the same as the English word `pinch`. You may consider it as meaning "in a crisis".

> 日本が **ピンチ** だ  
>  'Japan is **in a crisis**.'

  * The nominal used with weather is the topic when used with `は`, and the focus when used with `が`. `が` is not always the nominative case marker. It is sometimes the focus particle. Probably, it is making it difficult to you because English needs the expletive `it` when there is no meaningful subject, whereas Japanese does not need such thing.

> 明日 **は** 雨だ  
>  ' **As for** tomorrow, it will be raining.'
>
> 明日 **が** 雨だ  
>  ' **It is** tomorrow **that** it is going to rain.'
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Perhaps part of the solution is the dropping of words assumed from context?

明日は雨だ → 明日(の天気)は雨だ

You could consider this as a type of sentence known as "ウナギ文".

<http://www.sf.airnet.ne.jp/~ts/language/unagi.html> \- has a nice explanation
(in Japanese), exemplified by the exchange (ordering in a restaurant):

甲: 僕は天丼にするよ。

乙: 僕はうなぎだ。

Here the second speaker means that they are going to order the eel, not that
they are an eel. Where appropriate が can be used instead of は in this type of
sentence as well (example given at link).

For the other, I'd have a look at
<http://www.geocities.jp/niwasaburoo/02meisibunhosetu.html> .This link has
some description of (starting within section 2.3 from 「～が～だ」の形の記述文 and going
onwards) the type of thing you're looking for.

This is basically the website author's summary of things from some linguistic
texts, the one of interest is:

> 砂川有里子『文法と談話の接点』くろしお出版 2005

It describes sentences like (clipped) 家が火事だ as expressing the status (状態) at
the time of speaking, and goes on to split "コピュラ文" (copula sentences) into
various subtypes. The author of the book calls these particular sentences
現象描写文タイプ.

I think by comparison with 家が火事だ ( _the house is **on** fire_ rather than _the
house is a fire_ ) we could fit 日本がピンチだ quite neatly into that category.

Note: I couldn't even begin to guess at how widely accepted the work of this
particular person is, and there are clearly various different ways of
categorising コピュラ文 depending on which linguist you ask.
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Both 日本がピンチだ and 明日は雨だ are just usual subject-noun-copula sentences in
Japanese. The fact that they are not translated to subject-copula-noun
sentences in English has nothing to do with the structure of the Japanese
sentences per se.

Some people call nouns that are typically translated to adjectives “no-
adjectives.” While this notion may be convenient for learning purposes, you
should be aware that this notion is only defined in relation to other
languages to translate into. As far as the Japanese language itself is
concerned, so-called “no-adjectives” are just nouns. ([More on no-
adjectives](https://japanese.stackexchange.com/questions/2770/so-
called-%E3%81%AE-adjectives-how-does-%E3%81%AE-really-work/2771#2771).)
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No. 3 is the correct interpretation, to the best of my linguistic knowledge.
As indicated in the Wikipedia article, だ and its related forms (e.g. です) are
derived from で+ある (or some other existential verb) and similar things. Here ある
means to exist, and で is a general adverbial case marker, serving simply to
make a noun into an adverb. As this is a very vague case (in general Japanese
grammar is extremely vague in comparison to English), there are a wide array
of possible interpretations. The literal meaning of 日本がピンチだ would be "Japan
exists as/by/with/in [a] pinch", where "in" is the correct meaning, here (note
that with other verbs で can also mean "at", but with verbs of existence that's
expressed via the dative of possession).

For a cliche similar example, take あなたが好きです. Here, "with" is the closest
literal translation of で.
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