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# Tag Info

6

With CBC mode the initialization vector is referred to as IV, because it is not nonce. There are ways to construct nonce so that it does not meet the needs of CBC mode. Random IV is one generation choice which is usually fine. Nonce can also be a counter, which is not ok here. Definitions Nonce means number used once. IV means initialization vector. CBC ...

5

Under the assumption that $(K,\text{Msg})\to H_K(\text{Msg})$ is a secure MAC (be it HMAC or any other MAC), and $\text{Nonce}$ does not repeat and is of fixed size, both $H_K(\text{Msg}||\text{Nonce})$ and $H_K(\text{Nonce}||\text{Msg})$ are demonstrably secure, in the sense that an adversary not knowing $K$ can't distinguish either from random, even for ...

3

This is vulnerable to a length extension attack. Given a valid nonce/MAC, the nonce can be extended to forge a new valid nonce/MAC value. This is because $m_4$ is appended to the end inside the outer hash. How this affects you will depend on how you validate your nonce. But in general, this is not a secure construction. There's probably more things wrong ...

3

Let $2^m$ be the average message length in blocks. When using an independent random nonce for the whole 128-bit IV of each block, you would expect a collision after $2^{64}$ blocks, i.e. $2^{64-m}$ messages. (But you double the data size.) When using a 96-bit nonce and a 32-bit counter, you would expect a nonce collision after $2^{48}$ messages. This is ...

3

There is not much difference and in practice the terms are often used to mean the same thing. In this context however the Nonce does not have to keep to the random properties that the IV has. As explained in the paper: A probabilistic encryption scheme $C = \varepsilon^R_K (P)$ is an IV-based encryption scheme, syntactically, but we are suggesting that, ...

3

You looked on version 1.49 where OCB was not fully implemented as it seems. Actually OCB uses only 120 bit nonce, the other 8 bits are encoded as described in the RFC. Have a look at version 1.50. There OCB seems (nearly) fully implemented and an exception is raised, if the given nonce is longer than 15 bytes (source code line #158).

2

I'll give another answer in case you or someone else needs to work with that version of OCB and/or Bouncy Castle. My understanding of this check is that if the nonce is longer than 16 bytes, or the nonce is 16 bytes and the first bit of the first byte of the nonce is not 0 (assuming big endian), then an error is thrown. Do I understand this ...

2

An IV is an intial vector, which means it is an initial vector of data used when you start a chaining mode. It has no interesting properties of its own. If the IV is a nonce, that means it is a number used once (eg CTR mode). This means that (by changing the IV) we ensure that the process is never run on exactly the same input data (even if messages are ...

2

The once part inside of the nonce in CTR mode means effectively "once for this particular key". If you use a fresh key for each message (e.g. by encrypting it using public-key crypto or similar), you can use the same nonce for all the messages (or a size-zero nonce). The important part is that the combination of nonce and ctr-value (i.e. what is input into ...

2

Should the external nonce passed to GCM be authenticated separately when passing over network? No, that is not necessary; it is implicitly authenticated by GCM itself, pretty much as the AAD is also authenticated. That is, if someone in the middle modifies the nonce, then that will alter the authentication tag that the decryptor computes as a part of ...

2

Nonces must be unique but are not secret. Typically you send it alongside the ciphertext as a prefix. Note that with the asymmetric box, you must not use a nonce that you used in one direction in the opposite direction, since both directions use the same shared symmetric key. Reusing a nonce is a fatal mistake. It completely breaks the MAC and it leaks the ...

2

The treatment of nonces is the same for most stream ciphers, they only differ in length. In the case of Salsa20 the nonce is 64 bits, and there is a related cipher XSalsa20 which extends the nonce to 192 bits. The caller/application provides the nonce. Typically the protocol or file format you're implementing specifies how to treat the nonce. The essential ...

2

AES-CTR is very appropriate. Since a credit card number is 16 characters long, it can be encrypted using a single 128-bit block without any encoding. You will only need 1 block, and hence not require a block counter, just the nonce. Depending on the amount of card numbers being stored, you would only need to store a portion of the full nonce. A 32-bit ...

1

The standard approach is to have the sender pick his nonce (either randomly, or as a counter), and send it with the packet. The decryptor then knows what nonce to use to decrypt, because it's right there. Because nonces aren't assumed to be secret, this works.

1

If your nonce is 16 bytes, and your message pre-nonce is a multiple of 16 bytes (i.e. no padding is needed), sending the nonce in the clear opens you up to replay-ish attacks. Specifically, if an attacker captures one exchange with nonce $N$ and response $R$ (with $b$ blocks $R_1$ through $R_b$), and then impersonates the server and the client sends them ...

1

Well, you have it right in how nonces are used to make sure that the keys in different SSL sessions; this effectively prevents someone from taking an SSL record from one session, and injecting it into another -- because the keys aren't the same, it won't pass the integrity tests. However, that's not the only place we care about replay attacks; we can also ...

1

In the traditional sense, a nonce is a number that is only used once (with the same key). Though occasionally there are other requirements. For example, we may require that the nonce be unpredictable. For more on this I recommend you read this answer and this answer. Now, you don't go into enough detail on what you are doing to say whether or not you need ...

1

Passwords should use a password hashing function. Password hashing functions are different from basic cryptographic hashes, though they use cryptographic hashes as part of their construction. Password hashing functions must use salt. (Password hashing functions can also tune their time and/or memory usage, cryptographic hashes generally can't.) So for your ...

1

Yes, it's a bad idea. Take for instance the encryption part, and assume a stream cipher (as used in NaCL). The messages may be unique, but as the stream cipher requires a unique nonce you would loose all confidentiality! The easiest thing to do is to simply use a (random) nonce even if not strictly required. If you cannot do that because of bandwidth ...

1

All the security definitions I am aware of for a cryptographic hash functions remain the same, if you apply a 1:1 mapping before hashing. In other words, if $f$ and $g$ are each others inverse, and $h$ is a secure cryptographic hash, then $x \to h(f(x))$ is also a secure cryptographic hash according to any security definition, I know of. Under such ...

1

The attacker acts as a man-in-the-middle and can forward the clients nonce to the server and the servers nonce to the client. As a result both parties establish a connection with the attacker using the same nonces. Because of a weakness in the TLS protocol, the two connections can have the same key, and this key is known to the attacker. After resumption ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# 3.3.5: Data Tables

Position $$R_{AW}$$ $$R_{WB}$$ $$R_{AW} + R_{WB}$$
Fully CCW
1/4
1/2
3/4
Fully CW

Table $$\PageIndex{1}$$

Position $$V_{WB}$$ Open $$V_{WB}$$ 47k $$V_{WB}$$ 4.7k $$V_{WB}$$ 1k
Fully CCW
1/4
1/2
3/4
Fully CW

Table $$\PageIndex{2}$$

Position $$I_L$$ 1k $$I_L$$ 4.7k
Fully CCW
1/4
1/2
3/4
Fully CW

Table $$\PageIndex{3}$$

This page titled 3.3.5: Data Tables is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by James M. Fiore.

format: plain


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# All Questions

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text:
Happenings in the world of me
By gazbo (Wed Jan 10, 2007 at 12:42:19 AM EST) (all tags)
First off, I DEMAND AN MFC EXTENSION TILL MONDAY!  Reasons will be given within.

Update [2007-1-10 6:5:54 by gazbo]: Morrissey for Eurovision?

There is quite a lot of content inside.  This is possibly the first and last time I can say that.

Inside: New job!  Gigs!  Houses!  Love life!

New jobs:

Just before Christmas, I went for an interview with another Sheffield company, and received a job offer, which I have accepted.  I consider this as continuing the recent HuSi tradition of changing jobs, and also to make me less of a hypocrite when I mock other people for working in jobs they don't like.

It's not that there was anything wrong with my old job, but general ennui and the stress of having so much responsibility all to myself sucked.  It was hard not to play out in my head scenarios that could quite easily happen that would result in major crises that I would have to sort out on my own.  Even having one other person capable of helping would have eased the concern - misery shared and all that.

New job is still based on web interfaces, but heavily involving AJAX/SOAP and resulting in "cool" things (VOIP, IPTV).  The pay rise, better prospects and stability are all pluses too.

Favourite cheeky interview moment:

Interviewer: "We encourage working from home."
Gaz: "That's funny, so do I."

I start on the 15th.

Gigs:

I briefly mentioned the other day that I had formed a covers band, and that we were playing a gig last Friday.  Well, here are some more details.

I've wanted to do more music stuff for ages, and had had a drunken conversation with someone about how it would be cool just to form a cheesy covers band, just to get out there and play.  Zero artistic integrity, just a bit of fun and a little extra cash.  At the end of November he called me up and asked if I'd be happy playing bass rather than guitar, as he (vocalist) had found a guitarist (mutual friend, better guitarist than me) who in turn had found a drummer (who can keep time!  A drummer!  Who can keep time!)

Naturally I said yes, we picked 5 songs to learn that week, and went to rehearsal.  It was a bit messy, but there was a glimmer of hope.  This is when the vocalist mentioned he'd booked us a gig on Jan 5th playing at his friend's surprise birthday party, giving us about 5 weeks of elapsed time to learn a full set and practice enough to get tight.  We crammed in as many rehearsals as we possibly could, but that still only amounted to 5-6 rehearsals of 2-3 hours.  That's why I've not been around so much; free time has been spent learning basslines

After the last rehearsal, I was actually beginning to think we could do it.  We bought a lighting rig and associated gubbins, decided on a band outfit (jeans+t-shirt is fine for NME's darling credible indie bands, but not for covers bands) and set up at the venue - a pub in Chesterfield.

When everyone showed, I had time for a couple of nerve-settling pints, then we went backstage to put on the outfits and head out.  Nerves were on high terror alert, when I heard the opening riff to Hey Joe.  It happened so quickly I didn't have time to think about anything other than getting ready for my part.  Now Hey Joe is a very easy song for the bass, and thankfully nobody missed any cues for the different riffs.  It sounded great.

Quickly detune to drop-D, and straight into Fat Bottomed Girls.  This is the song I'd decided ahead of time I needed to get right, as it was so early on in the set and the first song where the bassline was really a major part of the song.  It went without hitch, and I was able to relax and get into it after that point.

The first set was 12 songs, ending on Love Machine, which went down really well.  Throughout the set people were politely listening but not much more.  Still, that's to be expected.  The reaction to Love Machine was better though.

Break for half an hour and have a couple more pints.  Nerves pretty much non-existent now.  This time when we came out, people seemed more into it.  We opened with that rock classic Since You've Been Gone, and people were shouting along.  By the end of the set (12 songs) the crowd were well and truly onside, shouting along, singing into the mic when the vocalist went running into the audience.  The only big mistake we made was in Stuck In The Middle where the vocalist went into the bridge when it should have been the guitar break.  We quickly switched to match him, but he just got confused.  Still, we eventually recovered in time for the next verse, so although a noticeable fuck-up, we didn't have to stop playing, which is about the worst thing that could happen.

We went off stage, and the crowd were literally roaring for more.  Now I know they were all friends and friends-of-friends, and so were onside from the start, but it was still a nice thing to hear.  So we switched hats (hats!) into the cowboy hats and did a 2 song encore of Sweet Home Alabama and finally Wanted Dead Or Alive.

After the gig, we got lots of positive comments, suggestions of people who'd want to book us, and general amazement when we said we'd only been together for 5 weeks.

I had a photographer friend take pics, so hopefully I should be able to post some in the near future.

Last night, I started to record my MFC entry (I wasn't leaving it till the last minute; for some reason I thought the deadline was Friday), when I realised that as backups I'd taken my XLR and TRS cables to the gig, and the vocalist still had them.  I'll be able to get them back at rehearsal on Saturday, so I humbly request an extension till Monday, as I'll be unable to record vocals otherwise :(

Houses:

My housemate's moving in with his girlfriend at the start of next month.  He tried to phone the landlord mid-December, but couldn't get through on any of three numbers.  We've not been able to get through since.  A friend said she may want to move in, but fucked around for ages.  She eventually said yes, but would only be able to move in May, and so phoned the landlord to see if they could work around that.  I didn't want to pay the full house rent for the months in-between, of course.

Naturally she couldn't get through either.

Fortunately I've found another friend who'll move in, and will move in from the start of Feb.  We've even managed to get in touch with the landlord after my housemate sent a speculative email to someone with their fairly distinctive name.  All I need to do now is get them on the phone and arrange a date for contract signing.  Huzzah!  Moving house is a pain - I still would like my next house move to be into a house I own.  Perhaps not next year, but the year after.

Love life:

Well, I've wanted out of my job for a while, and I got that sorted at the start of this year.  I've wanted to get more involved with music and performing, and I've now got that sorted.  The immediate issue of accommodation has now been sorted.  That leaves exactly one thing that I would like to change in my life: I require a Gazbette.

I've been quite happily single for several years, with just the occasional (alright, very occasional) one-night-stand.  This is due to a general wariness of getting into relationships for reasons that I shaln't go into.  But finally I think I can stand having a woman around.  Based on recent life-successes (see first paragraph of the section) I imagine I shall "score" within Q1.

Miscellany:

As if you needed telling, I am going to see Sugababes and Girls Aloud in the next few months.  I have decent seats for GA, but the Sugababes seats are literally A-M-A-Z-I-N-G.  So there.

Happenings in the world of me | 31 comments (31 topical, 0 hidden) | Trackback
Congratulations by anonimouse (4.00 / 2) #1 Wed Jan 10, 2007 at 12:46:49 AM EST
Incidentally, I heard GA do an acoustic version of the Amy Winehouse song "Rehab" on the radio and I have to admit I was impressed.

Girls come and go but a mortgage is for 25 years -- JtL
Yeah, I've got the MP3 by gazbo (4.00 / 1) #2 Wed Jan 10, 2007 at 01:08:16 AM EST
It was a good version - particularly impressed that they let Nicola take vocal duty for the first verse.

Maybe they felt that as it was radio they didn't have to worry about "the ugly one" getting screentime :(

I recommend always assuming 7th normal form where items in a text column are not allowed to rhyme.

[ Parent ]
Good work! by nebbish (4.00 / 2) #3 Wed Jan 10, 2007 at 01:17:19 AM EST
Was it you who chose "Love Machine" and "Since You've Been Gone" then? :) Inspired choices both!

Good luck on the lass front - I made a decision like that a while back, and, erm, well I hope it goes a bit better for you than it has for me.

--------

Sorry, I should clarify by gazbo (4.00 / 2) #5 Wed Jan 10, 2007 at 01:26:16 AM EST
That's "Since you've been gone" by Rainbow, not "Since u been gone" by Kelly Clarkson.

Actually, we did rehearse the Kelly Clarkson track, and Pink's "U + UR Hand", but sadly they didn't really work vocals-wise so we dropped them.

But yeah, Love Machine was one I pushed for.

I recommend always assuming 7th normal form where items in a text column are not allowed to rhyme.

[ Parent ]
Pink... by Breaker (4.00 / 2) #8 Wed Jan 10, 2007 at 01:52:43 AM EST
Is one of the most underrated vocalists IMHO.  CAB WTB bought a PS2 and SingStar for Crimbo, and the Pink songs are the hardest to sing on every edition of the game we have.

[ Parent ]
She's very good by nebbish (4.00 / 2) #11 Wed Jan 10, 2007 at 02:09:34 AM EST
Also, phwoar

--------

[ Parent ]
Chavvy, don't you think? by ambrosen (4.00 / 3) #14 Wed Jan 10, 2007 at 02:21:41 AM EST
Oh wait, I was expecting that to put you off.

[ Parent ]
There's chavvy and there's by dmg (2.00 / 0) #31 Sat Mar 01, 2008 at 04:52:37 AM EST
chavvy. Still whatever floats your boat I guess...
--
dmg - HuSi's most dimwitted overprivileged user.
[ Parent ]
Indeed. by Breaker (4.00 / 3) #16 Wed Jan 10, 2007 at 02:39:23 AM EST
Sadly she's married though, so unlucky!

[ Parent ]
Damn by nebbish (4.00 / 2) #18 Wed Jan 10, 2007 at 03:46:48 AM EST
I really thought I was in there as well

--------

[ Parent ]
Never mind. by Breaker (4.00 / 3) #20 Wed Jan 10, 2007 at 04:31:06 AM EST
You're still young, there's plenty more fish in the sea.  Britney's probably going to be needing some portioning soon, even if it'll be like throwing a sausage down a corridor.

[ Parent ]
I was surprised the other day. by ambrosen (4.00 / 2) #21 Wed Jan 10, 2007 at 04:56:02 AM EST
I was looking through a magazine someone had left lying around on a train, and the surprisingly pretty woman in the picture actually turned out to be our friend Brockley Spears, minus most of the stage makeup and with straight hair.

Sausage down a corridor? I heard she was saving herself.

[ Parent ]
Waiting for nebbish. by Breaker (4.00 / 2) #24 Wed Jan 10, 2007 at 05:23:23 AM EST
Although, having batched out 2 kids in as many years, it'd be like putting a needle through a wizards sleeve.

[ Parent ]
Did you see Jordan on Graham Norton ages ago? by nebbish (4.00 / 2) #26 Wed Jan 10, 2007 at 05:35:51 AM EST
He was interviewing a woman whose husband had a 12" penis. Jordan turns to her, totally deadpan, and says "You must have a right bucket."

Classic telly.

--------

[ Parent ]
Will have to look on youtube by Breaker (4.00 / 2) #29 Wed Jan 10, 2007 at 06:29:38 AM EST
NT

[ Parent ]
Good body by BadDoggie (4.00 / 2) #22 Wed Jan 10, 2007 at 04:59:41 AM EST
Great voice. Face like a drill sergeant except perhaps when wearing the spiky pink wig. I'm working on lyrics to a Boy's Response to U + UR Hand. I need someone to run that song through a vocal stripper for me.

woof.

OMG WE'RE FUCKED! -- duxup ?

[ Parent ]
I think she's gorgeous by nebbish (4.00 / 2) #23 Wed Jan 10, 2007 at 05:14:19 AM EST
She looks so dirty I don't even mind the freakishly short arms.

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[ Parent ]
Aye by Bob Abooey (4.00 / 3) #25 Wed Jan 10, 2007 at 05:33:21 AM EST
She's no super-model but she's certainly not an unattractive bird.

Warmest regards,

[ Parent ]
Supermodels? Pah! by nebbish (4.00 / 3) #27 Wed Jan 10, 2007 at 05:44:19 AM EST
Is there anyone less attractive than Kate Moss? (well obviously there is, but you know what I mean)

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[ Parent ]
Ah, OK by nebbish (4.00 / 2) #12 Wed Jan 10, 2007 at 02:10:40 AM EST
Dunno the Rainbow one. At least you tried the Kelly Clarkson one - and yeah, the vocals would be tricky.

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[ Parent ]
Typo? by Breaker (4.00 / 3) #4 Wed Jan 10, 2007 at 01:17:38 AM EST
I have decent seats for GA

Did you mean /I have decent seats for G-A-Y?

Heh by gazbo (4.00 / 2) #6 Wed Jan 10, 2007 at 01:27:31 AM EST
I'd take offence, except for the fact that Girls Aloud and G-A-Y are virtually synonymous.  They play there loads.

If I had a pair of assless chaps I'd go and see them.

I recommend always assuming 7th normal form where items in a text column are not allowed to rhyme.

[ Parent ]
That's strange by komet (4.00 / 2) #7 Wed Jan 10, 2007 at 01:44:51 AM EST
I was certain that Miss Gazbatine existed. Clearly my deductive powers are failing me.

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<ni> komet: You are functionally illiterate as regards trashy erotica.
You may have been confused by gazbo (4.00 / 1) #9 Wed Jan 10, 2007 at 01:53:15 AM EST
By talk of a young lady who attended the last LHuSi drinks I went to.  She would count under the category of one-night-stand.

Does it count as a one-night-stand if you've had two of them, separated by a few months?

I recommend always assuming 7th normal form where items in a text column are not allowed to rhyme.

[ Parent ]
Wait a second by ambrosen (4.00 / 3) #13 Wed Jan 10, 2007 at 02:19:56 AM EST
She came to a HuSi drinks and didn't immediately and forthwith cease all contact? Surely that woman's a keeper.

[ Parent ]
tecnicly by martingale (4.00 / 3) #15 Wed Jan 10, 2007 at 02:26:33 AM EST
that's two nightstands, but don't worry, nobody knows what they're for except women. It's like the mystery of matching duvet and pillow covers, really. I imagine the intelligent designer is a bit of a prankster, too.
--
$E(X_t|F_s) = X_s,\quad t > s$
[ Parent ]
Blimey by TPD (4.00 / 2) #10 Wed Jan 10, 2007 at 02:02:51 AM EST
It may have been a friendly crowd, but by the length of that set list still sounds like it was a pretty daunting first gig (as if I'd know).

Congratulations on it being such a great one and please let us know the dates of the world tour!

why sit, when you can sit and swivel with The Ab-SwivellerTM

You can have the extension to monday by cam (4.00 / 2) #17 Wed Jan 10, 2007 at 02:43:39 AM EST
what date is that in the UK?
Impressive... by Metatone (4.00 / 4) #19 Wed Jan 10, 2007 at 04:04:47 AM EST
Next project for you, organize a Curry night when nebbish is next in the area.

Plan! by nebbish (4.00 / 2) #28 Wed Jan 10, 2007 at 05:50:37 AM EST
I'm due a visit to Sheffield soon, once my sister's moved house properly (I seriously don't want to get caught out and end up spending a weekend decorating).

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text:
# proof of chain rule (several variables)

We first consider the case $m=1$ i.e. $G\colon I\to\mathbb{R}^{n}$ where $I\subset\mathbb{R}$ is a neighbourhood of a point $x_{0}\in\mathbb{R}$ and $F\colon U\subset\mathbb{R}^{n}\to\mathbb{R}$ is defined on a neighbourhood $U$ of $y_{0}=G(x_{0})$ such that $G(I)\subset U$. We suppose that both $G$ is differentiable at the point $x_{0}$ and $F$ is differentiable in $y_{0}$. We want to compute the derivative of the compound function $H(x)=F(G(x))$ at $x=x_{0}$.

By the definition of derivative (using Landau notation) we have

 $F(y_{0}+k)=F(y_{0})+DF(y_{0})k+o(|k|).$

Choose any $h\neq 0$ such that $x_{0}+h\in I$ and set $k=G(x_{0}+h)-G(x_{0})$ to obtain

 $\displaystyle\frac{H(x_{0}+h)-H(x_{0})}{h}$ $\displaystyle=\frac{F(G(x_{0}+h))-F(G(x_{0}))}{h}$ $\displaystyle=\frac{F(G(x_{0})+k)-F(G(x_{0}))}{h}=\frac{F(y_{0}+k)-F(y_{0})}{h}$ $\displaystyle=\frac{DF(y_{0})(G(x_{0}+h)-G(x_{0}))+o(|G(x_{0}+h)-G(x_{0})|)}{h}$ $\displaystyle=DF(y_{0})\frac{G(x_{0}+h)-G(x_{0})}{h}+\frac{o(|G(x_{0}+h)-G(x_{% 0})|)}{h}.$

Letting $h\to 0$ the first term of the sum converges to $DF(y_{0})G^{\prime}(x_{0})$ hence we want to prove that the second term converges to $0$. Indeed we have

 $\left|\frac{o(|G(x_{0}+h)-G(x_{0})|)}{h}\right|=\left|\frac{o(|G(x_{0}+h)-G(x_% {0})|)}{|G(x_{0}+h)-G(x_{0})|}\right|\cdot\left|\frac{G(x_{0}+h)-G(x_{0})}{h}% \right|.$

By the definition of $o(\cdot)$ the first fraction tends to $0$, while the second fraction tends to the absolute value of $G^{\prime}(x_{0})$. Thus the product tends to $0$, as needed.

Consider now the general case $G\colon V\subset\mathbb{R}^{m}\to U\subset\mathbb{R}^{n}$. Given $v\in\mathbb{R}^{m}$ we are going to compute the directional derivative

 $\frac{\partial F\circ G}{\partial v}(x_{0})=\frac{dF\circ g}{dt}(0)$

where $g(t)=G(x_{0}+tv)$ is a function of a single variable $t\in\mathbb{R}$. Thus we fall back to the previous case and we find that

 $\frac{\partial F\circ G}{\partial v}(x_{0})=DF(G(x_{0}))g^{\prime}(0).=DF(G(x_% {0}))\frac{\partial G}{\partial v}(x_{0})$

In particular when $v=e_{k}$ is the $k$-th coordinate vector, we find

 $g^{\prime}(0)=D_{x_{k}}F\circ G(x_{0})=DF(G(x_{0}))D_{x_{k}}G(x_{0})=\sum_{i=1% }^{n}D_{y_{i}}G(x_{0})D_{x_{k}}G^{i}(x_{0})$

which can be compactly written

 $DF\circ G(x_{0})=DF(G(x_{0}))DG(x_{0}).$
Title proof of chain rule (several variables) ProofOfChainRuleseveralVariables 2013-03-22 16:05:07 2013-03-22 16:05:07 paolini (1187) paolini (1187) 6 paolini (1187) Proof msc 26B12


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# Long-term measurements of cloud droplet concentrations and aerosol–cloud interactions in continental boundary layer clouds

## Abstract

The effects of aerosol on cloud droplet effective radius (Reff), cloud optical thickness and cloud droplet number concentration (N d) are analysed both from long-term direct ground-based in situ measurements conducted at the Puijo measurement station in Eastern Finland and from the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument onboard the Terra and Aqua satellites. The mean in situ N d during the period of study was 217 cm−3, while the MODIS-based N d was 171 cm−3. The absolute values, and the dependence of both N d observations on the measured aerosol number concentration in the accumulation mode (N acc), are quite similar. In both data sets N d is clearly dependent on N acc, for N acc values lower than approximately 450 cm−3. Also, the values of the aerosol–cloud-interaction parameter [ACI=(1/3)*d ln(N d)/d ln(N acc)] are quite similar for N acc<400 cm−3 with values of 0.16 and 0.14 from in situ and MODIS measurements, respectively. With higher N acc (>450 cm−3) N d increases only slowly. Similarly, the effect of aerosol on MODIS-retrieved R eff is visible only at low N acc values. In a sub set of data, the cloud and aerosol properties were measured simultaneously. For that data the comparison between MODIS-derived N d and directly measured N d, or the cloud droplet number concentration estimated from N acc values (N d,p), shows a correlation, which is greatly improved after careful screening using a ceilometer to make sure that only single cloud layers existed. This suggests that such determination of the number of cloud layers is very important when trying to match ground-based measurements to MODIS measurements.

Keywords:
How to Cite: Ahmad, I., Mielonen, T., Grosvenor, D.P., Portin, H.J., Arola, A., Mikkonen, S., Kühn, T., Leskinen, A., Joutsensaari, J., Komppula, M., Lehtinen, K.E.J., Laaksonen, A. and Romakkaniemi, S., 2013. Long-term measurements of cloud droplet concentrations and aerosol–cloud interactions in continental boundary layer clouds. Tellus B: Chemical and Physical Meteorology, 65(1), p.20138. DOI: http://doi.org/10.3402/tellusb.v65i0.20138
Published on 01 Jan 2013
Accepted on 27 Aug 2013            Submitted on 26 Nov 2012

## 1. Introduction

Atmospheric aerosols can have important direct radiative impacts, and indirect impacts through their effect on cloud microphysical properties. The total effect of aerosol particles on the global radiation budget is negative (IPCC, 2007), and thus they are partly counteracting the warming effect caused by the increased concentrations of greenhouse gases. However, due to highly varying concentrations and composition of aerosols, and feedbacks from the climate system onto aerosols and clouds, the total effect on the global radiation budget is not well known. Aerosols can affect the number concentration of cloud droplets (Nd) and thereby the impact that clouds have on radiative transfer in the atmosphere. However, this interaction is complex and it produces the largest single uncertainty in the estimates of Earth's radiation budget (IPCC, 2007). The interaction between aerosol and cloud, and cloud formation is not a trivial process; it depends on many factors including local weather parameters along with the chemical, optical and dynamical properties of aerosol and cloud (McFiggans et al., 2006; Romakkaniemi et al., 2006; Reutter et al., 2009).

Cloud droplet number concentration during the formation of cloud droplets is determined by the aerosol size distribution and chemical composition, the amount of condensable gases, the temperature and the updraft velocity of the air parcel entering the cloud (McFiggans et al., 2006). The first two define how many particles can maximally act as cloud condensation nuclei (CCN) at some supersaturation, and the last two depend on the local meteorology. Theoretically, an increase in the aerosol concentration should lead to an increase in Nd, which has been observed in several direct measurements of aerosol–cloud interactions (ACI) (e.g. Snider et al., 2003; Komppula et al., 2005; Meskhidze et al., 2005; Portin et al., 2009). If Nd increases throughout a cloud whilst holding the liquid water path (LWP) constant, the effective radius Reff of cloud droplets decreases, and this increases cloud optical thickness (COT) and cloud albedo (Twomey, 1977). This effect has been termed the first aerosol indirect effect. Furthermore, changes in Nd affect also drizzle formation so that increased Nd suppresses rain formation efficiency and ultimately enhances cloud fraction (Albrecht, 1989). This process has been termed the second indirect effect. However, due to the complexity of the processes of cloud and drizzle formation, it is not always trivial to observe these different effects as there are several feedbacks involved in the cloud systems (Stevens and Feingold, 2009).

Beyond direct in situ measurements, it is also possible to use remote sensing tools, such as satellite instruments (e.g. Nakajima et al., 2001) or ground-based instruments as deployed in the Atmospheric Radiation Measurement (ARM) programme (e.g. Dong et al., 2008), to study the aerosol effect on clouds. The difficulty with estimating ACI remotely by satellite is a lack of coincident information of both the cloud and the aerosol residing below the cloud. Thus, some studies have used ground-based data for aerosol properties and satellite data for cloud properties (e.g. Boers et al., 2006; Lihavainen et al., 2010; Janssen et al., 2011), and in some cases satellite retrievals have been validated with airborne measurements (e.g. Schüller et al., 2003; Painemal and Zuidema, 2011). Quite often different measurement methods give different cloud responses to aerosol perturbation and several reasons have been proposed for this difference (Lihavainen et al., 2010; McComiskey and Feingold, 2012).

In this article, we focus on ACIs observed both in situ and remotely. We use the measurement data from Puijo station in Finland together with cloud retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS) instruments onboard the Aqua and Terra satellites to assess the effect of aerosol concentration on Nd, COT and Reff. The relation between accumulation mode particle concentration (Nacc) from ground-based measurements and COT and Reff, from MODIS is studied to see how aerosol affects the optical and microphysical properties of low clouds. We also compare Nd and the aerosol–cloud interaction (ACI$=\frac{1}{3}\frac{dln\left({N}_{d}\right)}{dln\left({N}_{acc}\right)}$) (Feingold et al., 2001; McComiskey et al., 2009; Lihavainen et al., 2010) parameter values derived from satellite data to those from direct measurements. ACI parameters are a commonly used set of metrics that associate changes in Nd, Reff and COT to changes in aerosol burden, for example, Nacc or aerosol optical thickness. Here, we focus on the ACI that pertains to Nd. A significant amount of work has already been done regarding ACIs over ocean – for example, the VOCALS-Rex campaign (Wood et al., 2011) – but to our knowledge this is the first time that long-term ground-based aerosol and cloud droplet measurements have been compared to satellite retrievals for continental clouds.

## 2. Investigation site and data sets

### 2.1. Puijo measurement station

The measurement station at Puijo (62°54′34′′N, 27°39′19′′E) is on top of an observation tower located near the city of Kuopio. The measurements are carried out at a height of 306 m above sea level and 224 m above the surrounding lake level. The regional terrain consists mainly of lakes, small hills, forests, urban areas and rural areas. Overall, the surroundings of the measurement station are quite homogeneous with regard to aerosol sources, with only a few point sources of aerosol emissions. Since 2005/06, the station has been instrumented for continuous measurements of aerosols, cloud droplets, weather parameters and trace gases (Leskinen et al., 2009; Portin et al., 2009). Below, we give a brief description of the instruments used in our study, also listed in Table 1.

The aerosol size distribution is measured with a twin Differential Mobility Particle Sizer (DMPS) system which has a measurement range from 7 to 800 nm in diameter. The device is operated with two different inlets, which allows interstitial particles and those inside cloud droplets to be distinguished. For the direct measurements of cloud droplet number concentration and their size distribution a Cloud Droplet Probe (CDP, Droplet Measurement Technologies) is used. The device has a measurement range of 3–50 µm in diameter. In the present work, we have used one-hour-averaged data for Nacc and Nd for a large area (1°×1°) analysis, and 5 minutes Nd averages for a smaller area (5×5 km) analysis. In the latter, the shorter averaging time is used to match the satellite observation time as well as possible. In the comparison of cloud properties to aerosol data, a one-hour averaging time is used to get reliable statistics from the aerosol measurements as the DMPS system used takes 12 minutes for a single distribution scan. These data are compared with 1°×1° satellite data since data at this resolution are readily available as MODIS-level three data. The elevation of cloud layers is detected with a Ceilometer (Vaisala CT25 K) located at the Savilahti research station less than 2 km from Puijo and 219 m below the Puijo station measurement level.

Cloud events, defined as when the station is surrounded by clouds with high enough liquid water content (LWC>0.02 gm−3) for at least an hour, occur most frequently at Puijo in October when the top of the Puijo tower is inside cloud on more than 40% of days. Since June 2006, there have been 450 cloud events with a total 3000 hours of data. This data set was screened to remove rain events, discontinuous clouds, icing conditions and warm days (which cause an overheating of the CDP). After careful screening we are finally left with 414 cloud hours with reliable CDP data.

In an earlier study, Portin et al. (2009) analysed the dependence of Nd on aerosol properties for the same region. They showed that, in the cloud typical to Puijo station, Nd is dependent on Nacc, which they defined to be particles larger than 100 nm in diameter. It was found that a proxy cloud droplet number concentration (Nd,p) that matched the observed Nd well can be calculated with the following relation:

(1 )
${N}_{\text{d},\text{p}}=a×{\left({N}_{\text{acc}}\right)}^{b}$

Here the constants a and b were obtained by data fitting, with values of 30.13 and 0.36 respectively [these are updated values based on a larger data set than the one used by Portin et al. (2009)]. Using this equation we can estimate the droplet concentration of the clouds above the tower by assuming that the relation is representative for all boundary layer clouds observed close to the station.

### 2.2. Satellite data

We use collection 5.1 data from MODIS instruments onboard both the Aqua and Terra polar synchronous orbiting satellites. The primary quantities of interest were Reff and COT (MODIS ATBDT – cloud products; King et al., 1997; Platnick et al., 2003). We also used MODIS cloud-top temperature and cloud-top pressure data to ensure that the analysed clouds were low-level clouds. Further details on the data selection and filtering are provided in Section 3.

From the measured parameters, Reff and COT it is possible to estimate Nd as well as LWP (e.g. Wood and Hartmann, 2006; Bennartz, 2007) using

(2 )
${N}_{\text{d}}=\frac{{2}^{-2.5}}{k}{\left[COT\right]}^{3}{\left[\frac{LWP}{{C}_{\text{F}}}\right]}^{-2.5}{\left[\frac{3}{5}\pi Q\right]}^{-3}{\left[\frac{3}{4\pi {\rho }_{\text{w}}}\right]}^{-2}{C}_{\text{w}}$

and

(3 )
$LWP=\frac{5}{9}{\rho }_{\text{w}}\left(COT\right)\left({R}_{\text{eff}}\right)$

In this study all the quantities are averages over cloudy pixels only and therefore cloud fraction (CF) is 1. In eq. (2) k is the cube of the ratio between the volume mean radius (Rv) and the effective radius of droplets ($k={\left(\frac{{R}_{\text{v}}}{{R}_{\text{eff}}}\right)}^{3}$) with a given constant value of 0.67 (Martin et al., 1994) in our study, Q is the scattering efficiency whose value is approximately 2, and ρw is the density of water. For this calculation, it is assumed that LWC increases linearly with height from cloud base to cloud top (i.e. that the clouds are adiabatic). Cw is the ‘condensation rate’ for such a moist adiabatic ascent given by

(4 )
${c}_{\text{w}}=\frac{d{q}_{\text{L}}}{dz}=-\frac{d{q}_{\text{v}}}{dz}=\frac{{C}_{\text{p}}}{{L}_{\text{v}}}\left({\mathrm{\Gamma }}_{\text{d}}-{\mathrm{\Gamma }}_{\text{m}}\right)$

Here qv and qL are the water vapour and liquid mixing ratios; Lv is latent heat of condensation of water; Cp is the specific heat capacity of dry air at constant pressure; and Γd (=g/Cp, where g is the gravitational acceleration) and Γ m are the dry and moist adiabatic lapse rates, respectively. Cw depends mainly on temperature (T), although there is also some weak pressure (P) dependence. This equation can be derived through consideration of the conservation of moist static energy during moist adiabatic ascent, that is,

(5 )
${C}_{\text{p}}dT-\frac{{R}_{\text{a}}TdP}{P}+{L}_{\text{v}}d{q}_{\text{v}}=0$

Here Ra is the specific gas constant for dry air (=287.04 J kg−1 K−1). Then eq. (4) above can be obtained by dividing by dz, inserting the hydrostatic equation and recognising that the temperature change dT/dz will be the moist adiabatic lapse rate (Γm).

## 3. Cloud event selection

For the comparison of ground-based measurements and MODIS data, it is crucial that only a single layer of cloud exists. For some of our comparison single-layered cloud events were chosen using data from the Vaisala CT25 K Ceilometer and/or by limiting the MODIS-retrieved cloud-top pressure and temperature values. Using meteorological data from Puijo station and cloud-phase information from MODIS, we limited our study to non-precipitating low-level liquid clouds. A limit of satellite-derived Reff (3 µm ≤Reff≤30 µm) is also applied to make the selection of liquid-only cloud more reliable (Nakajima and King, 1990).

After screening out the possible ice phase and rain clouds, we further divided the satellite data used in this study into three categories:

1. For category I events the Puijo station (where the CDP instrument was located) was determined to be inside cloud by requiring that the LWC measured from the CDP was greater than 0.02 g m−3 for at least an hour and the visibility during that hour was less than 200 m. MODIS Level 2 collection 5.1 data were taken from the LAADS WEB website for both Aqua and Terra overpasses that occurred within the time window of a 5-minute averaged CDP data point. Nd was calculated using eq. (2) for the 25 1×1 km pixels centred on the measurement station to give a value averaged over 5×5 km. To avoid rain contamination, the maximum allowed rain intensity was limited to 0.2 mm hour−1. To help ensure that the cloud over the 5×5 km region was as close to plane parallel as possible it was required that at least 90% of the 25 pixels were designated as cloudy, clear or undetermined by the MODIS phase algorithm and did not suffer from sunglint. All of these pixels were then required to be assigned liquid cloud status, ensuring that at least 90% of the 5×5 km region was liquid cloud. In order for the data point to be used, it was required that at least 90% of the pixels that remained after the screening had successful Reff and COT retrievals and had cloud water path quality assurance flags of ‘very good confidence’ (the highest possible). The minimum cloud-top temperature within the 5×5 km area was also limited to be above 273°K to ensure that no ice could be present in the cloud. Reff retrievals made using the 2.1-µm band are used here for consistency with the Level-3 retrievals. However, using the 3.7-µm band retrievals produced similar results, perhaps due to the efforts to only include clouds that were homogeneous in terms of cloud fraction. Category I data is used for the direct comparison of cloud number concentration obtained from the in situ CDP measurements and from MODIS retrievals. Thirteen events were available for this comparison. Five-minute averages of Nd data from the CDP were used and were matched closely to the MODIS overpass time in order to account for the likelihood of the rapid time evolution of the cloud fields and to help prevent spatial variability and local effects from biasing the comparison. 5×5 km data points with a MODIS Nd standard deviation greater than 150 cm−3 were also removed.
2. Category II events consist of cloud layers with base height less than 800 m above Savilahti station (likely boundary layer cloud). The maximum rain intensity (obtained from the weather station) is limited to 0.2 mm hour−1 and cloud-top temperature (from MODIS) was limited to more than 265 K. The same limit for cloud-top temperature was used in the International Cloud Climatology Project (ISCCP) classification for low-level cloud (Cairns, 1995). In this category, the cloud was directly above Puijo in the boundary layer, and thus we assume that the aerosol measured at Puijo represents well the aerosol at cloud base. Thus we calculate Nd,p from the DMPS Nacc value using eq. (1). For comparison, MODIS 5×5 km Level 2 collection 5.1 data were used to calculate a MODIS Nd in the same manner as for category I. The difference here is that the lifting of the restriction that the Puijo station is inside cloud allows more data points to be considered. In this category, 62 events were available for analysis over the observation period (2006–2011).
3. Category III consists of cloud layers with cloud-top temperature and pressure more than 265 K and 780 hPa, respectively. In this category, the region is cloudy, but we do not know whether there was cloud directly above the Puijo station. Nevertheless, we assume that the one-hour-averaged Nacc that approximately corresponds to the times of the local MODIS (Aqua and Terra) overpasses is representative of the regional boundary layer aerosol over the observational region. We calculate Nd,p using eq. (1). This was compared to Nd values calculated using the mean Reff and COT from MODIS (Aqua and Terra) 1°×1° Level 3 collection 5.1 data, which was taken from the GES DISC Giovanni website (http://disc.sci.gsfc.nasa.gov/giovanni/). A total of 1058 MODIS overpasses were recorded since 2000, out of which 481 remained with coincident Nacc data over the period of 2006–2011.

## 4. Results

The satellite-derived Reff and COT from the category III data have average values of 11.6±2.9 µm and 13.2±8 µm, respectively, for the observation period 2006–2011. These values are comparable to those presented by Janssen et al. (2011) derived over a 2°×2° area above the Hyytiälä measurement station 200 km southwest of Puijo station. Similar results can be expected because both areas are quite comparable in terms of aerosol sources. The average Nd calculated from eq. (2) is 171 cm−3. This value is higher than the range 40–100 cm−3 obtained by Janssen et al. (2011) and at the upper end of the range 45.2 cm−3 (clean pixels) – 132 cm−3 (polluted pixels) obtained by Sporre et al. (2011) for a cleaner area. One possible reason for the difference is the different method used to derive Nd, as Janssen et al. (2011) used models from Boers et al. (2006). We also tested the method used in Sporre et al. (2011) obtaining an average Nd of 156 cm−3. For comparison, the average Nd from the CDP data is 217 cm−3, which is quite close to the MODIS-measured Nd.

To study the dependence of Nd on aerosol particles we have analysed the number concentration (Nacc) of particles larger than 100 nm in diameter at the Puijo measurement station. We also tested different minimum sizes between 80 and 150 nm for the definition of Nacc particles, and found that the following results are insensitive to these choices. In Fig. 1 we compare Nacc to MODIS-retrieved Reff, COT and Nd for the category III data. The line indicates a running mean of 30 data points to make trends more readable for eye. We can see that at low values of Nacc both Reff and Nd are clearly dependent on Nacc. At high values this is not the case. The calculated Spearman correlation between MODIS-retrieved Nd and Nacc for values less than 400 cm−3 is 0.38, and the ACI parameter is 0.14 when calculated from the linear fit between Nacc values of 70 cm−3 and 400 cm−3. For larger Nacc, there is no correlation and the average Nd remains almost constant at around 215 cm−3. For COT, we found no dependence on Nacc, which is interesting given the observed dependence of Nd on Nacc since it would be expected that for a fixed adiabatic cloud thickness (H) COT would increase proportionally with Nd1/3. However, COT is also proportional to H5/3 in such an adiabatic cloud – that is, a cloud with the same assumptions as those used in deriving eqs. (2) and (3) – and thus is much more sensitive to changes in H than to changes in Nd. Such changes in H are likely to occur due to the influence of local meteorological changes, thus making it more difficult to see the effect of Nd and aerosol on COT in the rather noisy data.

Fig. 1

The effect of accumulation mode particle number concentration on effective radii, (a), cloud droplet number concentration (b) and cloud optical thickness (COT) (c) retrieved from Moderate Resolution Imaging Spectroradiometer (MODIS) during 2006–2011. Red lines are the 30-point running mean. Our category III data are used here.

Similarly to Fig. 1, in Fig. 2 we show how the in situ measured Nd from the CDP depends on the Nacc. In this case, the Spearman correlation is 0.63 for Nacc less than 400 cm−3. As expected, the correlation is higher in direct measurements as the cloud droplets and aerosols are measured in the same place. The ACI value calculated from these data was 0.16 (and 0.12 using eq. (1) with fitted parameters) on average for Nacc lower than 400 cm−3. Both ACI values are very similar to those reported by Garrett et al. (2004) who documented ACI values in the range of 0.13–0.19 using ground-based measurements of low-level liquid clouds observed near Barrow Alaska. Lihavainen et al. (2010) reported measurements conducted in a cleaner area in Northern Finland, where ACI values of 0.2 to 0.3 were obtained from ground-based measurements and approximately 0.1 from satellite measurements.

Fig. 2

Cloud droplet number concentration as a function of accumulation mode aerosol number concentration from Puijo station between 2006 and 2011. The red line is a 30-point running mean and the green line is a fit based on eq. (1).

Both data sets show that Nd is only weakly, or not at all dependent on Nacc at high aerosol loadings. The reason for this finding is the complex dynamics of cloud formation. In the type of clouds observed at Puijo, the updraft velocities are quite low, and in such conditions Nd is typically only weakly dependent on aerosol particle concentration above some threshold concentration (Reutter et al., 2009; Romakkaniemi et al., 2012). When comparing the in situ and MODIS data sets, we can see some differences. For example, for the in situ measurements the running mean of Nd is still increasing at high aerosol loading, but this is not the case in the MODIS-derived Nd. This might be because for the in situ measurements there are no small Nd values with high aerosol loading, but in MODIS-derived data such data points do exist. As we are correlating local aerosol concentrations with cloud properties determined from a larger area, it is always possible that some local source might have increased the measured aerosol concentration above that of the observational area, or that some local aerosol are not effective as CCN.

In Fig. 3 we show Nd retrieved from MODIS as a function of that directly measured in the cloud (category I, red circles) or estimated from aerosol measurements at Puijo station (category II, blue circles). In the case of direct measurements we have also included the standard deviation of the observed Nd over the 5×5 km area. It can be seen that the in situ CDP and MODIS Nd values are slightly anti-correlated, with a correlation coefficient of −0.07 (but with 95% confidence limits ranging from −0.6 to +0.5) and an RMSE value of 118.9 cm−3. There is a tendency for the MODIS Nd values to be clustered around the same value, whereas corresponding CDP Nd values can reach somewhat higher values. The mean value from the in situ measurements (271 cm−3) is higher than the MODIS-retrieved one (209 cm−3). Here, we also tested different spatial averaging from MODIS and temporal averaging from CDP measurements, but the agreement between retrievals was not improved.

Fig. 3

Moderate Resolution Imaging Spectroradiometer (MODIS)-retrieved Nd for categories I and II versus Nd or Nd,p from Puijo station, as measured from the Cloud Droplet Probe (CDP) (red, Nd) and estimated from eq. 1 (blue, Nd,p), respectively. The vertical bars for the category I MODIS data show the standard deviations over the 5×5 km areas combined with the Moderate Resolution Imaging Spectroradiometer (MODIS) measurement error. The horizontal bars show the CDP standard deviation over the 5-minute averaging period.

However, given the low number of data points the correlation could easily have occurred by chance. Indeed the 95% confidence limits of the correlation coefficient cover a very wide range suggesting that this is a strong possibility. When the data were further filtered to remove multiple-layer cloud using ceilometer, much better agreement was found (r=0.44 with upper and lower 95% confidence limits of −0.47 and 0.9, respectively). However, only seven data points remained making firm conclusions difficult.

When category II data, where the in situ Nd is estimated (Nd,p) from the measured Nacc using eq. (1) and thus the restriction that the Puijo station is in cloud is lifted, are compared to MODIS-derived Nd, the correlation is also good. But again only when the data are filtered using the ceilometer data to make sure single boundary layer cloud (cloud base height less than 800 m) existed, although there are still some points that are far from the one-to-one line. The RMSE and correlation coefficient values in this case are 99.6 cm−3 and 0.65 (the 95% confidence limits are 0.47 and 0.78), respectively. The results from the category I and category II data suggest that the height of the cloud under consideration and whether it is within the boundary layer, which is likely well mixed, is very important for matching ground measurements to satellite measurements. Additionally, the good match between MODIS and the Nd,p suggests that the MODIS retrievals themselves are not be the sole cause of the CDP discrepancy.

To further analyse the differences in cloud droplet number concentrations derived from different measurements the frequency distributions for Nd are shown in Fig. 4. Here we use data from satellite and in situ measurements from the months when the CDP was operating. Thus the distributions presented include all the data from the previous figures, and also non-coincident data from the same time period (2006–2011). It can be seen that the MODIS-derived Nd is peaked at smaller values with a median of 148 cm−3. The in situ measurement gives a slightly broader distribution of Nd values with a median of 216 cm−3.

Fig. 4

Probability distributions of cloud droplet number concentrations from in situ Cloud Droplet Probe (CDP) measurements and from Moderate Resolution Imaging Spectroradiometer (MODIS) for the time 2006–2011. Both coincident and non-coincident CDP measurements and MODIS data are used here. Blue line denotes CDP and green line denotes MODIS.

Our in situ measurements of cloud droplet number concentration are in good agreement with several other measurements conducted in continental stratus clouds (e.g. Leaitch et al., 1992; Dong et al., 2005; Komppula et al., 2005). The mean MODIS-retrieved Nd in this study is in reasonable agreement with direct measurements and most likely the Nd retrieval method used in Janssen et al. (2011) gave too low values. However, as can be seen from eqs. (2) and (3), there are several assumptions that need to be made to calculate Nd using MODIS-retrieved COT and Reff. The calculated Nd is more sensitive to Reff than to COT and Painemal and Zuidema (2011) concluded that the MODIS-retrieved Reff systematically exceeded the in situ measured value by 15–20%. As the power of Reff in eq. (2) is −2.5, the decrease of Reff by 15–20% (when taken alone) would increase retrieved Nd by a factor of 1.5–1.75 and bring the median MODIS value from Fig. 4 into good agreement with that from the in situ data. However, Painemal and Zuidema (2011) also found that, for their assumptions of k=0.8 and of adiabatic clouds in the calculation of Nd, the overestimate of Reff by MODIS was compensated by an underestimate of the k factor (the actual value observed was ~0.88) and the observation that the clouds were fairly sub-abiabatic. They found a measured effective Cw of 0.7 times the adiabatic Cw value. However, whether the parameters observed by Painemal and Zuidema (2011) in the South-East Pacific are relevant to the Puijo region is an open question.

Thus, there are potentially large uncertainties in the estimate of Nd, which may explain some of the differences between the in situ and MODIS Nd seen here. However, it is likely that constant systematic biases in the parameters just mentioned would lead to a constant relative difference between the Nd values. This suggests that non-systematic biases were occurring in either of these parameters, the MODIS measurements or the in situ CDP measurements (or a combination). Another possible reason for discrepancy is the assumption that Nd is constant throughout the cloud, which is implicit in the Nd calculation and is also necessary to make sure that the tower observed value is not different from that higher up in the cloud. Aircraft measurements have suggested that this is generally the case in clouds with bases above ground level (Martin et al., 1994; Miles et al., 2000; Wood, 2005), although this has not been well tested for clouds near the ground. An alternative explanation is that MODIS is more likely to sample higher clouds than the surface-based CDP and thus it is possible that Nd at such elevated levels is lower than Nd near the surface due to a lower aerosol concentration.

## 5. Conclusions

We have compared in situ measured cloud microphysical properties to those retrieved from the MODIS instruments onboard the Aqua and Terra satellites. We have used ground-based measurements from the Puijo measurement station to study the dependence of cloud droplet number concentration (Nd) and cloud droplet effective radius (Reff) on the number concentration of aerosol particles in the accumulation mode (Nacc). As far as we know, this is the first time that Nd analysed from long-term ground-based in situ measurements have been compared to satellite observations. We have found that in situ measurements showed a clear correlation between Nd and Nacc for Nacc concentrations less than 400 cm−3. A similar but weaker correlation was seen from the remote sensing data. The ACI value (0.16) from in situ measurements was almost similar the ACI value (0.14) calculated from the MODIS-retrieved Nd. This is valid for Nd values lower than 250 cm−3 and Nacc<400 cm−3. Above this value Nd increased only slightly (in situ measurements), or levelled off at around 215 cm−3 (MODIS). Mean (median) Nd from in situ measurements was a factor of 1.2 (1.4) higher than the MODIS-derived value. However, due to several uncertainties in the method used to calculate Nd it is difficult to say if this difference was caused by, for example, a systematic error in the retrieval of Reff.

As shown, our Nd values are clearly higher than, for example, those measured in the study of Janssen et al. (2011), even though the study areas partly overlap. This suggests that the satellite-retrieved Nd is strongly dependent on the retrieval method. This must be taken into account for example when MODIS-retrieved Nd is used for model validation purposes.

The results presented above initially showed fairly poor agreement between the in situ CDP Nd and that measured from satellite. However, when the data were filtered using ceilometer data to make sure that the clouds were within the boundary layer and that only a single layer of cloud existed, the agreement was much improved. Unfortunately, the amount of data for the direct comparison was very low, and so it is difficult to draw some conclusion. Good agreement was obtained when the satellite data was compared to an Nd,p value calculated using the aerosol measurements, but again only if the data were first filtered using ceilometer measurements. This suggests that such height determination is very important when trying to match ground-based measurements to satellite measurements.

## 6. Acknowledgments

This work was supported by Maj and Tor Nessling Foundation, Academy of Finland (through the Centre of Excellence programme, project no 1118615), and by the strategic funding of the University of Eastern Finland. Analysis and visualisations used in this study were produced with MODIS level 2 and level 3 atmosphere data taken from LAADS WEB (Goddard Space Flight Centre) and Giovanni online data system, developed and maintained by the NASA GES DISC.

## References

1. Albrecht B. A . Aerosols, cloud microphysics, and fractional cloudiness . Science . 1989 ; 245 : 1227 – 1230 .

2. Bennartz R . Global assessment of marine boundary layer cloud droplet number concentration from satellite . J. Geophys. Res . 2007 ; 112 : 02201 .

3. Boers R , Acarreta J. R , Gras J. L . Satellite monitoring of the first indirect aerosol effect: retrieval of the droplet concentration of water clouds . J. Geophys. Res . 2006 ; 111 : 22208 .

4. Cairns B. C . Diurnal variations of cloud from ISCCP data . Atmos. Res . 1995 ; 37 : 133 – 246 .

5. Dong X , Minnis P , Xi B . A climatology of midlatitude continental clouds from the ARM SGP Central Facility: part I: low-level cloud macrophysical, microphysical, and radiative properties . J. Clim . 2005 ; 18 : 1391 .

6. Dong X , Minnis P , Xi B , Sun-Mack S , Chen Y . Comparison of CERESMODIS stratus cloud properties with ground-based measurements at the DOE ARM Southern Great Plains site . J. Geophys. Res . 2008 ; 113 : 03204 .

7. Feingold G , Remer L. A , Ramaprasad J , Kaufman Y. J . Analysis of smoke impact on clouds in Brazilian biomass burning regions: an extension of Twomey's approach . J. Geophys. Res . 2001 ; 106 : 22907 .

8. Garrett T. J , Zhao C , Dong X , Mace G. G , Hobbs P. V . Effects of varying aerosol regimes on low-level Arctic stratus . Geophys. Res. Lett . 2004 ; 31 : 17105 .

9. International Panel on Climate Change (IPCC Report) . Solomon S , Qin D , Manning M , Chen Z , Marquis M , co-authors . Climate change (2007), the physical science basis . Contribution of working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change .

10. King M. D , Tsay S , Platnick S. E , Wang M , Liou K . Cloud retrieval algorithms for MODIS: optical thickness, effective particle radius and thermodynamic phase, MODIS Algorithm Theoretical Basis Document (ATBD-MOD-05) . 1997 ; NASA . Online at: http://modis-atmos.gsfc.nasa.gov/_docs/atbd_mod05.pdf (Accessed Aug 2012) .

11. Komppula M , Lihavainen H , Kerminen V.-M , Kulmala M , Viisanen Y . Measurements of cloud droplet activation of aerosol particles at a clean subarctic background site . J. Geophys. Res . 2005 ; 110 : 06204 .

12. Leaitch W. R , Isaac G. A , Strapp J. W , Banic C. M , Wiebe H. A . The relationship between cloud droplet number concentrations and anthropogenic pollution: observations and climatic implications . J. Geophys. Res . 1992 ; 97 : 2463 – 2474 .

13. Leskinen A , Portin H , Komppula M , Miettinen P , Arola A , co-authors . Overview of the research activities and results at the Puijo semi-urban measurement station . Boreal Environ. Res . 2009 ; 14 : 576 – 590 .

14. Lihavainen H , Kerminen V.-M , Remer L. A . Aerosol–cloud interaction determined both in situ and satellite data over a northern high-latitude site . Atmos. Chem. Phys . 2010 ; 10 : 10987 – 10995 .

15. Martin G. M , Johnson D. W , Spice A . The measurement and parameterization of effective radius of droplets in the warm stratocumulus clouds . J. Atmos. Sci . 1994 ; 51 : 1823 – 1842 .

16. McComiskey A , Feingold G . The scale problem in quantifying aerosol indirect effects . Atmos. Chem. Phys . 2012 ; 12 : 1031 – 1049 .

17. McComiskey A , Feingold G , Frisch A. S , Turner D. D , Miller M. A , co-authors . An assessment of aerosol–cloud interaction in marine stratus clouds based on surface remote sensing . J. Geophys. Res . 2009 ; 114 : D09203 .

18. McFiggans G , Artaxo P , Baltensperger U , Coe H , Facchini M. C , co-authors . The effect of physical and chemical aerosol properties on warm cloud droplet activation . Atmos. Chem. Phys . 2006 ; 6 : 2593 – 2649 .

19. Meskhidze N , Nenes A , Conant W. C , Seinfeld J. H . Evaluation of a new cloud droplet activation parameterization with in situ data from crystal-face and cstripe . J. Geophys. Res . 2005 ; 110 : 16202 .

20. Miles N. L , Verlinde J , Clothiaux E. E . Cloud droplet size distributions in low-level stratiform clouds . J. Atmos. Sci . 2000 ; 57 : 295 – 311 .

21. Nakajima T. Y , Higurashi A , Kawamoto K , Penner J. E . A possible correlation between satellite-derived cloud and aerosol microphysical parameters . Geophys. Res. Lett . 2001 ; 28 ( 7 ): 1171 – 1174 .

22. Nakajima T. Y , King M. D . Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements, Part I: theory . J. Atmos. Sci . 1990 ; 47 : 1878 – 1893 .

23. Painemal D , Zuidema P . Assessment of MODIS cloud effective radius and optical thickness retrievals over the Southeast Pacific with VOCALS-REx in situ measurements . J. Geophys. Res . 2011 ; 116 : 24206 .

24. Platnick S , King M. D , Ackerman S. A , Menzel W. P , Baum B. A , co-authors . The MODIS cloud products: algorithms and examples from Terra . IEEE Trans. Geosci. Remote Sens . 2003 ; 41 ( 2 ): 459 .

25. Portin J. H , Komppula M , Leskinen A. P , Romakkaniemi S , Laaksonen A , co-authors . Overview of the aerosol-cloud interaction at the Puijo semi-urban measurement station . Boreal Environ. Res . 2009 ; 14 : 641 – 653 .

26. Reutter P , Su H , Trentmann J , Simmel M , Rose D , co-authors . Aerosol- and updraft-limited regimes of cloud droplet formation: influence of particle number, size and hygroscopicity on the activation of cloud condensation nuclei (CCN) . Atmos. Chem. Phys . 2009 ; 9 : 7067 – 7080 .

27. Romakkaniemi S , Arola A , Kokkola H , Birmili W , Tuch T. M , co-authors . Effect of aerosol size distribution changes on AOD, CCN and cloud droplet number concentration: case studies from Erfurt and Melpitz, Germany . J. Geophys. Res . 2012 ; 117 : D07202 .

28. Romakkaniemi S , Kokkola H , Lehtinen K. E. J , Laaksonen A . The influence of nitric acid on the cloud processing of aerosol particles . Atmos. Chem. Phys . 2006 ; 6 : 1627 .

29. Schüller L , Brenguier J. L , Pawlowska H . Retrieval of microphysical, geometrical, and radiative properties of marine stratocumulus from remote sensing . J. Geophys. Res . 2003 ; 108 : 8631 .

30. Snider J. R , Guibert S , Brenguier J.-L , Putaud J.-P . Aerosol activation in marine stratocumulus clouds: 2. Köhler and parcel theory closure studies . J. Geophys. Res . 2003 ; 108 : 15 .

31. Sporre M , Glantz P , Tunved P , Swietlicki E , Kulmala M , co-authors . A study of the indirect aerosol effect on subarctic marine liquid low-level clouds using MODIS cloud data and ground-based aerosol measurements . Atmos. Res . 2011 ; 116 : 56 – 66 .

32. Stevens B , Feingold G . Untangling aerosol effects on clouds and precipitation in a buffered system . Nature . 2009 ; 461 : 607 – 613 .

33. Twomey S . The influence of pollution on the shortwave albedo of clouds . J. Atmos. Sci . 1977 ; 34 : 1149 – 1152 .

34. Wood R . Drizzle in stratiform boundary layer clouds. Part I: vertical and horizontal structure . J. Atmos. Sci . 2005 ; 62 : 3011 – 3033 .

35. Wood R , Hartmann D. L . Spatial variability of liquid water path in marine low cloud: the importance of mesoscale cellular convection . J. Clim . 2006 ; 19 : 1748 – 1764 .

36. Wood R , Mechoso C. R , Bretherton C. S , Weller R. A , Huebert B , co-authors . The VAMOS Ocean-Cloud Atmosphere-Land Study Regional Experiment (VOCALS-REx): goals, platforms, and field operations . Atmos. Chem. Phys . 2011 ; 11 : 627 .


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# 5.4.2: Field on the Axis of a Ring

[ "article:topic", "authorname:tatumj" ]

Before starting, one can obtain a qualitative idea of how the field on the axis of a ring varies with distance from the centre of the ring. Thus, the field at the centre of the ring will be zero, by symmetry. It will also be zero at an infinite distance along the axis. At other places it will not be zero; in other words, the field will first increase, then decrease, as we move along the axis. There will be some distance along the axis at which the field is greatest. We’ll want to know where this is, and what is its maximum value.

$$\text{FIGURE V.1}$$

Figure V.1 shows a ring of mass $$M$$, radius $$a$$. The problem is to calculate the strength of the gravitational field at $$\text{P}$$. We start by considering a small element of the ring of mass $$δM$$. The contribution of this element to the field is

$\frac{GδM}{a^2+z^2},$

directed from $$\text{P}$$ towards $$δM$$. This can be resolved into a component along the axis (directed to the centre of the ring) and a component at right angles to this. When the contributions to all elements around the circumference of the ring are added, the latter component will, by symmetry, be zero. The component along the axis of the ring is

$\frac{GδM}{a^2+z^2} \cos θ = \frac{GδM}{a^2+z^2} \cdot \frac{z}{\sqrt{a^2 + z^2}} = \frac{GδM z}{\left( a^2 + z^2 \right)^{3/2}}.$

On adding up the contributions of all elements around the circumference of the ring, we find, for the gravitational field at $$\text{P}$$

$g = \frac{GMz}{\left( a^2 + z^2 \right)^{3/2}} \label{5.4.4} \tag{5.4.4}$

directed towards the centre of the ring. This has the property, as expected, of being zero at the centre of the ring and at an infinite distance along the axis. If we express $$z$$ in units of $$a$$, and $$g$$ in units of $$GM/a^2$$, this becomes

$g = \frac{z}{\left( 1 + z^2 \right)^{3/2}}. \label{5.4.5} \tag{5.4.5}$

This is illustrated in figure V.2.

Excercise:    Show that the field reaches its greatest value of $$\frac{\sqrt{12} GM}{9a^2} = \frac{0.385 GM}{a^2}$$ where $$z = a/√2 = 0.707a$$. Show that the field has half this maximum value where $$z = 0.2047a$$ and $$z = 1.896a$$.

$$\text{FIGURE V.2}$$


text:
# If $\tan(x_1) \cdots\tan(x_n)=1$ for acute $x_i$, then does it follow that $\cos(x_1)+\cdots+\cos(x_n) \leq n\sqrt{2}/2$?

It is easily seen that if $$x,y\in[0,\pi/2)$$ satisfy $$\tan(x)\tan(y)=1$$, then $$\cos(x)+\cos(y)\le\sqrt 2$$

A much more delicate fact is that if $$\tan(x)\tan(y)\tan(z)=1$$ (while $$0\le x,y,z<\pi/2$$), then $$\cos(x)+\cos(y)+\cos(z)\le \frac{3\sqrt 2}2$$ I can prove this, but the proof is a little complicated; can anybody suggest a nice, simple proof?

As a generalization, suppose that $$n\ge 4$$, $$x_1,\dotsc,x_n\in[0,\pi/2)$$, and $$\tan(x_1)\dotsb\tan(x_n)=1$$. Does it follow that $$\cos(x_1)+\dotsb+\cos(x_n) \le \frac{n\sqrt 2}2?$$

• Could you provide your proofs? – Dr. Mathva Feb 13 at 12:15
• You can try first to rewrite the cosine using cos^2=1/(1+tg^2). Than a=tg(x),b=tg(y), c=tg(x), the costraint becomes abc=1 with a,b,c>0. Than go on with e^a'=a, e^b'=b and e^c'=c, so that the constrain becomes additive, a'+b'+c'=0. Using convexity finally of f(x)=\sqrt(1/(1+e^2x) should do the job. It is just an idea, but I do not have time to see if it really works :) – Thomas Feb 13 at 12:17
• @Thomas I checked your idea. It does not work. – Michael Rozenberg Feb 13 at 12:20
• @Dr.Mathva: $n=2$ is almost immediate, for $n=3$ see Michael Rozenberg's solution below. – W-t-P Feb 13 at 12:25
• @Michael Rozenberg . Yes that function is not convex. It would have been too easy :) Thx a lot for checking! – Thomas Feb 13 at 12:31

For three variables we ca use C-S:

Let $$\tan{x}=\sqrt{\frac{b}{a}},$$ $$\tan{y}=\sqrt{\frac{c}{b}},$$ where $$a$$, $$b$$ and $$c$$ are positives.

Thus, $$\tan{z}=\sqrt{\frac{a}{c}}$$ and by C-S we obtain: $$\sum_{cyc}\cos{x}=\sum_{cyc}\frac{1}{\sqrt{1+\tan^2x}}=\sum_{cyc}\sqrt{\frac{a}{a+b}}\leq$$ $$\leq\sqrt{\sum_{cyc}\frac{a}{(a+b)(a+c)}\sum_{cyc}(a+c)}\leq\frac{3}{\sqrt2},$$ where the last inequality it's just $$\sum_{cyc}c(a-b)^2\geq0.$$

The generalization is wrong for all $$n\geq4$$.

Try $$x_1=x_2=...=x_{n-1}\rightarrow0^+$$ and $$x_n\rightarrow\frac{\pi}{2}^-$$

• This is exactly the solution for $n=3$ I had in mind (actually, learned it from you). – W-t-P Feb 13 at 12:24
• @W-t-P Your generalization is very interesting. I'll think about this later because I am very very busy this week. Sorry. – Michael Rozenberg Feb 13 at 12:28

After the substitutions described in the comments ($$cos(x_i) \rightarrow \sqrt{\frac{1}{1+tg^2(x_i)})}$$ and $$e^{a_i}=tg(x_i)$$, one sees that needs to maximize:

$$f(x)=\sum_{i=1}^{n} \sqrt{\frac{1}{1+e^{2x_i}})}$$

subject to:

$$\sum_i x_i=0$$

over the domain $$x \in R^n$$.

This sounds like a job for Lagrange mulipliers but I was not able to solve the system. For the moment I observe that the conjecture is equivalent to having maximum for $$x_i=0$$ and I want to show that this is not the case for large $$n$$. Indeed $$f(0,...0)=n\sqrt{2}/2$$, but $$f(-m,-m,...,-m,(n-1)m)\rightarrow (n-1)$$ for large m. The second term dominates for large $$n$$ and therefore the thesis cannot be true.

• Too bad I just saw that the counter example had already been proposed editing the answer by Michel Rosenberg... Well since I did the effort to write my answer I will leave it... – Thomas Feb 14 at 8:17
• Anyway this "symmetry breaking" as a function of n (dimensionality) is very nice... Looks a bit "physical" :) – Thomas Feb 14 at 8:29