Just to add to MattM's answer, here is how the Kodokan Throwing Techniques video illustrates the difference in the sweeps in the two throws: Yama-arashi same side eri grip off-balancing uke forwards standing directly in front of uke, sweeping upwards leg parallel to uke's legs almost fully in contact Harai-goshi standard collar grip / under-arm grip off-balancing uke sidewards standing across uke, sweeping across leg leg almost perpendicular to uke's legs in contact only at one point
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Yama arashi examples from Kodokan Judo Throwing Techniques show sweep In Toshiro Daigo's book Kodokan Judo Throwing Techniques published by Kodansha International in 2005, which I consider the best canonical source for throw classification
, there are two examples of yama arashi and two examples of throws that are explicitly not yama arashi .
All of these have tori's right leg very close to or in contact with uke's right leg.
This is Daigo's description of what makes yama arashi distinct from page 77:
Tori places his right calf over uke's right shin and wraps the tip of his right foot around uke's right ankle.
In the unique feature of this technique, tori then sweeps up with his right leg placed tight against uke's right leg.
This sweeping technique is different from the method employed in harai-goshi.
Both examples of yama arashi involve a leg sweep.
The examples are distinguished by tori's right arm position when tori's left hand is on uke's right sleeve.
tori's right arm on uke's right chest area tori's right elbow inserted under uke's right armpit (as in a morote seoi nage )
The examples of throws that are not
yama arashi have the same hand positions as the yama arashi examples, but with no mention of a sweep: tai otoshi where tori's hands are on one side of uke, on the sleeve and the lapel seoi nage with the same hand position as the tai otoshi but with tori's elbow into uke's armpit as in a morote seoi nage
My interpretation is that yama arashi requires a sweep under current Kodokan classification.
If there is no sweep, it is not a yama arashi .
In the video referred to by Sjana , the throw shown by Mifune would not be considered a yama arashi but a tai otoshi under Daigo's description.
Based on Daigo's explicit counterexample, I suspect the throw in the video may have been considered a yama arashi by some practitioners before, but should not be now.
Kodokan judo throw classification is messy, changes with time, and is not always understandable.
Unless you are working for an official exam, it is frequently not worth expending the effort to understand the minutiae.
martialarts.stackexchange.com/users/5961/mattm
The name is different with the leg sweep.
Historically, yama arashi is catagorized as a "hand throw".
The fundamental impulse comes from the arms as you lift your uke over your hip.
You do unbalance them through the hip, but the primary force is through the arms.
Compare that to harai goshi , which is classified as a "hip throw" and the primary impulse is being delivered via the hip, while the arms are being used to unbalance.
In practice, of course, the throws are kind of on a continuum based on where you split the most effort, but the pure forms are "hand" and "hip" respectively.
You might further compare with osotogari , which is a similar technique, but is a "foot throw" with a clear sweep.
martialarts.stackexchange.com/users/1780/Macaco Branco
There is a very good video on youtube which explains the difference between Yama Arashi and Harai Goshi.
Harai Goshi doesn't require you to lift your opponent up.
Whereas the Yama Arashi uses an arm throw with your opponent in a lifted position.
From there your leg is only used to keep their foot from being planted forward.
Where the Harai Goshi uses the leg rotate your opponent, rather than using your arms.
https://www.youtube.com/watch?v=fM2Kjq4Ec2c
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Roughly, Yama-arashi could be classified as a variation of harai-goshi on the same basis that eri-seoi-nage is a variation of morote-seoi-nage.
Attention: according to IJF rules once you have both hands on the same side, you just have few seconds to throw with Yama-arashi, otherwise you will receive a shido.
martialarts.stackexchange.com/users/7114/kevin ternet
I have seen the throw  yama-arashi  done in multiple ways:   10 dan Mifune  performs it without sweeping uke's leg.  In the  Kodokan training video  tori performs  yama arashi  with an obvious leg sweep (seemingly as a close variation of  harai-goshi ).  I was shown the the throw by a 6 dan in Judo with no leg sweep.    What is the deal here? Why the differences in teaching a technique with the same name? Why is the name not different by adding the leg sweep?
What is the difference between Yama-arashi and Harai-goshi?
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martialarts.stackexchange.com/users/8964/Logikal
If there is no sweep, it is not a  yama arashi.  Historically,  yama arashi  is catagorized as a "hand throw". Harai Goshi doesn't require you to lift your opponent up. Hara Goshi does not require lifting the  opponent and has no sweep, whilst  Yamarashi is categorized as a  hand throw.
Hara Goshi doesn’t require you to lift your opponent and has no sweep. Yamarashi is categorized as a  hand throw.