text:
This Sylmar 'Voice' competitor was…
This Sylmar 'Voice' competitor was a wrestler before pursuing music
Delaney Silvernell performs "In My Blood" by Shawn Mendes during blind auditions for Season 15 of NBC's "The Voice."- (Photo by Tyler Golden/NBC)
PUBLISHED: October 14, 2018 at 7:24 am | UPDATED: October 14, 2018 at 7:24 am
Seconds into her blind audition for Season 15 of NBC's "The Voice," Sylmar singer Delaney Silvernell heard applause suddenly ring out.
Silvernell, 21, said she recalled thinking, "Oh! I must be doing something cool."
#NewProfilePic and probably the best one so far eh?? #teamkelly #thevoice pic.twitter.com/vXzkf1Kfz5
— Delaney Silvernell (@im_delaney_) October 2, 2018
"Then I looked up," she said, "and I saw Kelly looking at me."
With "The Voice" coach Kelly Clarkson bobbing along to her rendition of "In My Blood" by Shawn Mendes, Silvernell had secured a spot on Team Kelly, which goes into battle on Monday, Oct. 15.
Silvernell, the only local competitor, couldn't have planned it any better.
"'Miss Independent' by Kelly Clarkson was the first song I ever learned; I idolized Kelly Clarkson," said Silvernell, adding that was in the second grade in her hometown of Queensbury, New York where her family has been involved in the sport of wrestling for generations.
She also wrestled before giving it up in middle school to focus on singing.
"I was really young when I decided, That's what I wanted to do," she said. "But it was uncharted territory for my family and also a little intimidating. They wanted to be supportive, but they didn't know what the best course of action was."
By age 12, her parents found her a voice coach and she went on to major in songwriting at the Berklee School of Music. And after graduating last year she decided to move to Los Angeles instead of following her friends to New York.
"I wanted to reinvent myself," said Silvernell, who hopes "The Voice" can help.
At the very least, she'll come away with a satin red jacket emblazoned with Team Kelly on the back and KC 15 (for Kelly Clarkson, Season 15) on the front.
As she put it, "It's probably the coolest thing I own."
When: 8 p.m. Mondays and Tuesdays
Where: NBC
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It was reported that the wreckage of the aircraft which disappeared over the English Channel in January, carrying the Cardiff City striker and his pilot was found on Sunday.
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I have had 5 miscarriages. I had my first fresh cycle round of IVF in 2011 when I was 35. It was successful and I saw a heartbeat on the screen at 6w and 7w and then at 8w there was no heartbeat. I had to have a procedure to remove everything (called and ERPC a couple of weeks later). 2 years later, when I was 37 I had another round of IVF and it was a mirror image of what happened the first time except this time there was no procedure. The doctors told me to allow 'nature to take its course' and for my body to naturally miscarry (which was a traumatic event) Then in 2014 when I was 38 I fell pregnant naturally. Again, I saw a heartbeat on the screen at 6w and 7 w and at 8w the heartbeat was gone. In the summer of 2016 I had a biochemical pregnancy. In Jan of 2017 I fell pregnant naturally. 2 weeks later I miscarried. I am now 41.
Did you feel you had support from family, friends your GP and the Hospital?
GP didn't support me until I'd had 3 miscarriages as they said that they couldn't properly investigate me until I'd had 3. Unfortunately for me this took me to the age of 40. I was sent to St Mary's Recurrent Miscarriage Clinic who said I had an abnormal thromboelastogram and that I had 'sticky blood'. They said that upon falling pregnant again I would simply need heparin and aspirin and I should be able to retain a pregnancy successfully. I had 2 further miscarriages after that. The follow up care from the medical professional has been awful. I guess they are just doing their job and don't really 'care'. My friends have always been amazing. Most of them have had children in the time that I have been trying and always told me very early on when they found out that they were pregnant – to soften the blow I guess.
What was the hardest thing for you at the time?
The hardest thing for me about it is not having an absolute answer as to why this thing, that is so easy for other people, is so hard for me. I know that in order to get answers I have to push and push. I know that the problem is with me however is it my eggs or is it my environment? I have taken it upon myself to go for the Natural Killer Cell testing at a hospital in Coventry (where the two top professors in this subject work) on 20th March at a cost of £550. I feel desperate and very sad about the situation. It's a very lonely position to be in, especially as those who love me really don't know what to say or do so it's just awkward for them.
I don't think I have healed. I have good days and bad days and try not to spend time thinking about how sorry I feel for myself. I spend a lot of time with my niece, nephew and Godson although it doesn't do much to heal my aching heart. I'm hoping that I will become a mum at some point – whether it's my own baby, or I adopt or foster. I think by not giving up, I make myself feel better.
Where are you on your journey now?
I feel that generally the NHS don't care about women like me. I feel that IVF should be a free service for 3 rounds and shouldn't be discriminatory. That testing for recurrent miscarriages should be offered as standard. I could have easily fallen into a deep depression over my situation, which would have cost the NHS money anyway. In other countries IVF is free or heavily subsidised. Everything about my journey has been difficult and it really shouldn't be. I work hard and pay my taxes so would have liked to have been looked after a bit better.
It was 7 years ago, I was 42, I was very happy to be pregnant and there was a big gap after my son and daughter but I knew it would work. I felt good, I was at a healthy point in my life.
When I first went to the Dr and told her I thought I was pregnant she said. "Oh well, women of your age it doesn't always work" I had done the test and thought I'm pregnant and I thought it was a mean thing to say. It crossed my mind that I should be getting more help as in previous pregnancies I was prescribed progesterone. She did give me a light dose of tablets. One day I felt some pain and felt a bit uncomfortable but nothing happened.
I was booked in for the 12 week scan which ended up being at 14 weeks and that was when the horror began. You go along and just know that something was wrong. In hindsight just before that I didn't feel as pregnant as prior to that. It was a hell of a shock and they were just looking and looking and looking and then they step out of the room and don't say anything to you. I'd had two healthy pregnancies before this and I just knew something was wrong…. It was just like the world had stopped at that moment.
After that it was pretty matter of fact – I was given the option of letting it happen naturally or a termination via a D&C. We opted to go private because I wanted it to be over as quickly as possible. However it was a very cold experience, all very matter of fact, done and dusted and then we were home and had to get on with it. I was asked what religion I was and if I wanted a burial – which shocked me as I just wasn't expecting that and it was so final.
Because I was just at 14 weeks, I was waiting for my scan before I told any of my family and friends, so it was a double blow that I had to tell them I was pregnant and now I wasn't. My mother called me a dark-horse, when all I wanted was some support and kindness.
I felt incredibly low, just "done in" and wanted to go to bed and never get up. My thoughts were suicidal. A friend who had early miscarriages was there for me when she could be and my husband I felt wasn't there for me at the time. I guess it was his way of coping to go out and leave me. My friend told my husband to get a doctor as she was very worried about me. The doctor came and immediately prescribed anti-depressants. This is what got me through – it numbed me and it was the only way I coped with life as a mother, a wife and a business woman. I carried on regardless and didn't speak of it again.
It wasn't until I heard you speak of your story that it brought it up for me again and now I feel that it's really helped to talk about it.
Has it affected your relationship with your husband and family?
Yes, I have since divorced and I really feel that was a pivotal point in our marriage when I realised he wasn't there for me.
My family didn't really understand, they were a bit cross that I hadn't told them the news I was pregnant in the first place, but I was superstitious and wanted to wait for the all clear after the scan. It wasn't spoken of again, I guess they thought I was coping and was ok.
It was when I got home, my suicidal thoughts, and the most traumatic thing was the scan.
The anti-depressants were the only thing that got me through and with some family traumas and my divorce I stayed on them until 2 weeks ago.
In hindsight would you have done anything differently?
Yes I would have sought help and talked about it either counselling or psychotherapy. It would have been really helpful to have had someone who had also been through it who understood.
Looking back have you found any meaning?
No, no meaning at all – but maybe I wasn't on my right path and it happened to highlight something for me to be a better person, otherwise I would have just been Mum, Wife & Businesswoman. Sometimes I get guilty feelings about having had a cigarette that one day when I felt bad. But I know that wasn't the reason. I think it made me stronger "I survived that". It planted some seeds that all was not well in my relationship – It wasn't the fairytale I thought it was.
I help you transform the trauma of losing your baby.


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Archive	Monthly Archives: November 2016
Puzzle Play and Child Development
/ Children
/ By	isliplibrary
The Best Kind of Learning is the Kind that's Wrapped Up in a Game.
While puzzles may seem like mere child's play, researchers have discovered they are an important link in the cognitive development of young children. Researchers at the University of Chicago have found that children who play with puzzles between the ages of two and four develop better spatial and other important skills, including mathematical skills.
With almost 400 circulating puzzles available, the Islip Public Library is one of only a few libraries in Suffolk County that permits patrons to take up to five puzzles home for a week. Recently, the Library has expanded its puzzle lending policy to include out-of-district patrons, and renewals for an additional week for each puzzle checked out. In the supportive environment of their own homes, parents and children can play together to enrich a toddler's learning capacity and other skills.
Additionally, there are more than 30 puzzles, from large-knob puzzles, to mix-and-match puzzles, to alphabet puzzles, and more, suitable for infants to kindergarteners for in-library play.
Psychologist Susan Levine, a professor at the University of Chicago and a leading expert on mathematics development in young children, believes that children who play with puzzles perform better on tasks that are indicators of future math skills. Parents who interact with their children during puzzle play further advance their children's future skills in several different areas. Very young children can be guided by their parents, while pre-schoolers and kindergarteners enjoy putting puzzles together on their own.
Levine is the lead author of an academic paper studying children and parents as they engaged in puzzle play. For the research, 53 parent-child pairs from diverse socioeconomic backgrounds participated in a study in which researchers video-recorded parent-child interactions for 90-minute sessions that occurred every four months between children aged 26 to 46 months old. Researchers determined that children who play with puzzles at home improve future math-related skills.
Other researchers have learned that puzzle play increases fine motor development in young children, skills that eventually help with handwriting, drawing and even playing a musical instrument. Manipulating both large and small pieces of puzzles into the proper place requires eye-hand coordination, a most important skill in child development.
Larger puzzles can be solved by two or more children, which in turn helps increase a child's social skills. Solving puzzles as a group helps youngsters understand patience, teamwork and taking turns, according to an article on Huffington Post online. The Islip Public Library circulating puzzle collection includes a number of intricate layered puzzles, which are ideally suited for group puzzle solving.
Research indicates that toddlers who work on puzzles often progress to jigsaw puzzles and larger floor-sized puzzles, which are available online and in toy stores at nominal cost. Crossword, game and logic puzzles challenge children and aid in their cognitive development. Some good sources of puzzles for all ages is Amazon (https://www.amazon.com/Best-Sellers-Toys-Games-Toddler-Puzzles/zgbs/toys-and-games/196611011), and Lakeshore Education Products (http://products.lakeshorelearning.com/learning/Wooden-Puzzles). The website for Fine Wooden toys offers Melissa & Doug puzzles, which make a variety of interesting puzzles (http://www.finewoodentoys.com/). Very high-quality handmade solid wood puzzles are available for sale at Kidpuzzles (http://www.kidpuzzles.com/).
Best Sellers in Toddler Puzzles - amazon.com
Discover the best Toddler Puzzles in Best Sellers. Find the top 100 most popular items in Amazon Toys & Games Best Sellers.
Fine Wooden Toys and Wooden Puzzles for Babies and Young ...
www.finewoodentoys.com
At Fine Wooden Toys we offer a beautiful and unique selection of hundreds of top-quality educational and developmental wooden toys, blocks and puzzles for babies ...
Wooden Puzzles - Lakeshore Learning Materials
products.lakeshorelearning.com
Wooden Puzzles found in: Lakeshore Preschool Puzzle Library with Rack, Wild Animals Puzzle Set, Big Knob First Puzzle Set, Chunky Puzzles Set...
Perhaps most importantly, puzzle play gives beginning learners a huge sense of accomplishment. Watching toddlers put the last piece into a puzzle is a wonderful sight. Puzzles are fun, educational toys that challenge young minds, and the Islip Public Library's puzzle collection helps its youngest patrons achieve important skills.
The Picky Eater
/ Adults	Uncategorized
/ By	Michele Ferrari
When my son was younger, I thought of him as my "prickly pear." On certain topics he had to be approached in a particular way to elicit cooperation. Looking back now, I believe that he had sensitivity issues that were glimpsed but not fully comprehended, i.e. not liking the beach because of the feel of the sand, anxiety when traveling (worried about getting lost) and a most definite aversion to most foods. Almost every situation could be negotiated with reason, comedy and positive peer pressure -he has a great sense of humor and a very social aspect to his personality, as well as an older brother who could sometimes talk him into doing fun family things. Eating poorly, however, because of its very life and death nature, was an area that for me was fraught with dangerous consequences.
A great part of nurturing, of course, involves food and the role it plays in the life of a healthy and thriving child. However, when our best efforts are thwarted by what seems to be mere stubbornness on the part of our offspring, it can be frustrating and hurtful, especially if we see resistance as a reflection of our parenting skills. It's easy to see how food then becomes so inextricably entwined with our emotions, usually love. If we can take a step back and look at what else may be happening with the child, we may be able to employ strategies for de-escalating the conflicts arising at mealtimes. You might find that being a little flexible helps lessen the stress and signals to the child that you are paying attention to what their aversion to certain foods may signify. Of course, the issue is a complex one with many variations on the theme, depending upon the individual child and family, but, if it is any consolation, know that you are not alone!
With that thought in mind, there are a plethora of resources with ideas for making those food-related interactions more bearable.
Starting with the toddler/preschool set, the Baby Center gives some straightforward advice: http://www.babycenter.com/0_7-new-strategies-for-feeding-a-picky-eater_10360322.bc
Although playing with our food was once forbidden, who hasn't tried that old "airplane" ploy, or something like it, when spoon feeding a baby? Parenting Magazine's website has fun recipes and tips for enticing the pickiest of eaters: http://www.parenting.com/recipes/picky-eaters
The Mayo Clinic has some words of wisdom from nutrition experts:
http://www.mayoclinic.org/healthy-lifestyle/childrens-health/in-depth/childrens-health/art-20044948
Pediatric Occupational Therapist and mother Alisha Grogan has a website that addresses sensory issues and more: http://yourkidstable.com/
For the results of a scholarly study on what researchers term "selective eating":
http://www.webmd.com/children/news/20150803/kids-picky-eating#1
Also, it's a great idea for kids to be involved in the planning of their meals, including shopping and planting vegetable gardens:
http://smarteatingforkids.com/2013/05/13/helping-picky-eaters-grow-into-healthy-eaters/
This is just a small sampling of the many online sites a desperate parent could find useful. Some also cite additional sources for further information on the topic.
Of course, here at the library, there are print resources for parents
(please check our catalog):
Weelicious Lunches : Think Outside the Lunchbox with More than 160 Happier Meals
by Catherine McCord
Food Fights : Winning the Nutritional Challenges of Parenthood Armed with Insight, Humor and a Bottle of Ketchup by Laura A. Jana & Jennifer Shu
As well as downloadable eBooks: Helping your Child with Extreme Picky Eating: a Step-by-Step Guide for Overcoming Selective Eating, Food Aversion, and Feeding Disorders by Katja Rowell and Jenny McGlothlin
And Audiobooks: Say Goodbye to Fussy Eating by Lynda Hudson
For the kiddos themselves, we have many non-fiction, highly-illustrated books that talk about nutrition and the importance of eating healthy foods:
Taste Something New: Giving Different Foods a Try by Jennifer Boothroyd
Eat a Rainbow: Healthy Foods by Susan Temple Kesselring
Cool Eating: Healthy and Fun Ways to Eat Right by Alex Kuskowski
Eating Healthy from the Healthy Plates series by Valerie Bodden
Here are some colorful and instructional gardening books to let them understand where certain foods come from and to help grow them:
Garden to Table: a Kid's Guide to Planting, Growing, and Preparing Food by Katherine Hengel
Blue Potatoes, Orange Tomatoes by Rosalind Creasy
The Nitty-Gritty Gardening Book by Kari Cornell
Grow Your Own Monsters by Nicola Davies
Cookbooks geared toward children with simple and fun recipes so that they help create what goes in their mouths:
Kids cook 1-2-3: recipes for young chefs using only 3 ingredients by Rozanne Gold
Salad people and more real recipes : a new cookbook for preschoolers & up by Mollie Katzen
Cool kids cook: fresh & fit by Kid Chef Eliana with Dianne de Las Casas
Fiction authors use picky eaters as characters in their usually humorous stories:
Tales for Very Picky Eaters by Josh Schneider
The Luck of the Loch Ness Monster: a Tale of Picky Eating by A. W. Flaherty
D.W. the Picky Eater by Marc Brown
Gregory, the Terrible Eater by Mitchell Sharmat
Hopefully, with all this advice now available, parents and children will be able to reach a peaceful accord and make mealtime a much more harmonious occasion. As for my son, he has reached the ripe old age of 25 by eating snow-covered trees (cauliflower), olives (pitted, slip on fingers), elbow macaroni (the only acceptable shape), and spoonfuls of peanut butter, and, even though you would have to tie him down to make him eat cheese (involves mold, utterly disgusting) he still manages to be one of the healthiest people I know!
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redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
\label{sec_intro}

Water ice is an abundant material in the solar system. In the inner
solar system, it plays an important role on Earth and on Mars, where
it forms a cryosphere as an active, dynamic part of the respective
climate system. Approximately 10\% of the land surface of the
present-day Earth, or $14.6\times{}10^6\,\mathrm{km^2}$, are covered
by ice sheets, ice caps and glaciers, the total volume of which is
approximately $28.7\times{}10^6\,\mathrm{km^3}$ \cite{church_etal_01}.
By far the largest single ice body is the Antarctic ice sheet, which
alone contains about 90\% of this ice. All of these terrestrial ice
bodies are subject to gravity-driven glacial flow with typical
velocities of tens to hundreds of meters per year.  Further components
of the terrestrial cryosphere are the floating ice shelves and sea
ice, seasonal snow and ground ice (permafrost).

The ice sheets, ice shelves, ice caps and glaciers have formed by
accumulated snowfall over centuries, millenia and more. Over these
time-scales, the Earth's climate has experienced significant changes
known as glacial-interglacial cycles, which are driven by periodic
changes of the orbital parameters obliquity (axial tilt), eccentricity
and precession (``Milankovitch cycles''). The last 800,000 years have
been characterized by a strong dominance of the 100,000-year
eccentricity cycle \cite{raymo_etal_97}, with a sequence of shorter
interglacials (warm periods) like the current Holocene and longer
glacials (ice ages). At the last glacial maximum approximately 20,000
years ago, large parts of north America and Eurasia were covered by
ice sheets which no longer exist, and the global ice volume was about
three times larger than at present.

On Mars, the polar ice caps are one of the most prominent surface
features. The seasonal caps, which can extend down to latitudes of
approximately $55\degN$/S, consist of only some ten centimeters of
CO$_2$ snow which sublimes into the atmosphere during the respective
spring season. By contrast, the smaller residual caps poleward of
approximately $80\degN$/S survive the summer seasons, and they are
underlain by massive topographic structures, which are known as the
polar layered deposits \cite{thomas_etal_92}. The residual caps and
the underlying layered deposits are considered to be geomorphological
units and shall be referred to as the north- and south-polar cap
(NPC/SPC), respectively.  The Mars Orbiter Laser Altimeter (MOLA)
measurements of the Mars Global Surveyor (MGS) spacecraft have
provided a very precise mapping of the surface topographies of the
polar caps \cite{smith_etal_99a, zuber_etal_98}.  Combined with the
estimated cap margins and equilibrated ground topographies discussed
later (Sect.~\ref{sect_mpc_sico}), this yields for the NPC a volume of
about $1.2\times{}10^6\,\mathrm{km^3}$ and an area of
$1.1\times{}10^6\,\mathrm{km^2}$, and for the SPC a volume of
$1.8\times{}10^6\,\mathrm{km^3}$ and an area of
$1.7\times{}10^6\,\mathrm{km^2}$. Due to isostatic deflection of the
underlying lithosphere, the real volumes may be up to 30\% larger.  In
any case, the NPC and SPC are the largest known water reservoirs on
Mars. Further constituents may be dust, CO$_2$ ice and CO$_2$
clathrate hydrate.  Comparable or even larger amounts of water may be
stored as permafrost in the ground \cite{clifford_etal_00}; an idea
which was corroborated by the spectrometric detection of mid- and
high-latitude subsurface layers enriched in hydrogen, interpreted as
ground ice \cite{boynton_etal_02}.

Similar to the situation on Earth, the Martian polar caps are active
components of the climate system which interact with the atmosphere
thermally, orographically and by condensation and sublimation
processes of water vapour. Their present topographies are the result
of the climatic history over at least the last millions of years,
which were probably characterized by climate cycles as a consequence
of strong, quasi-periodic variations of the orbital parameters
obliquity, eccentricity and precession on time-scales of
$10^5$--$10^6$ years \cite{laskar_etal_04}. This idea is supported by
the light-dark layered deposits of the polar caps, which are exposed
in the scarps and troughs in the ice surface and close to the margins,
and which indicate a strongly varying dust content of the ice due to
varying atmospheric conditions in the past.
Further, Head et~al.\ \cite{head_etal_03} suggested that Mars
underwent ``ice ages'' during periods of high obliquity
like that from about 2.1 to 0.4 million years ago (with obliquity
maxima of ${}\approx{}35^\circ$). These ice ages are
supposedly characterized by warmer polar climates, enhanced mass loss
of the polar caps due to sublimation and the formation of meters-thick
ice deposits equatorward to approximately $30\degN$/S.

For the Moon and Mercury, which are both devoid of any significant
atmosphere and therefore subject to intensive solar radiation,
water ice may be cold-trapped inside permanently shadowed craters
at the poles, which is possible due to the very small tilts of the
rotational axes. Some evidence for this idea was provided by the
space-probe Lunar Prospector, which detected hydrogen in the
vicinity of the lunar poles by neutron spectrometry
\cite{feldman_etal_01}, and, for the case of Mercury, by
terrestrial radar mapping which revealed a highly reflective
region on the north pole \cite{slade_etal_92}.

In the outer solar system (beyond the asteroid belt), owing to the
very low temperatures, evaporation of water ice at the surface of
planetary bodies becomes so small that this substance survives for
time-spans comparable to the age of the solar system
\cite{watson_etal_63}. Therefore, while in the inner solar system
the geological evolution of the planets is dominated by rock
material, the history of a large number of bodies in the outer
solar system is dominated by water ice and other moderately
volatile substances. Of course, this statement does not hold for
the giant planets, which do not have a solid surface and are made
up of highly volatile substances, mainly hydrogen and helium.
Thus, it refers to the large moons of the Jovian, Saturnian,
Uranian and Neptunian systems, Pluto and smaller moons, asteroids
and planetary ring particles. Of particular interest are the
Jovian moon Europa, for which there is strong evidence for an
outer icy shell underlain by a deep ocean \cite{kargel_etal_00},
and the Saturnian moon Titan, for which the findings of the
recently landed space-probe Huygens suggest a solid surface made
up mainly of water ice and a methane hydrosphere with rainfall,
rivers and seas (see http://huygens.esa.int).

In this review paper, the focus will be on \emph{flowing} water ice,
which can be driven either by gravity forces or by thermal convection.
The layout is as follows. In Sect.~\ref{sect_ice_rheology}, the
rheology of polycrystalline aggregates of water ice is discussed,
including the effects of different creep mechanisms, partial melting
and impurities (dust), and suitable flow laws (stress-strain-rate
relations) are given.  This allows the formulation of a closed set of
model equations for the thermo-mechanically coupled problem of ice
flow, which is done in Sect.~\ref{sect_model_ice_flow}. Applications
to the Martian polar caps are presented in Sect.~\ref{sect_mpc}. New
results are shown for their large-scale dynamics under assumed
steady-state conditions (Sect.~\ref{sect_mpc_sico}), and the local
influence of the prominent scarps and troughs in the surface is
treated (Sect.~\ref{sect_mpc_scarp}). Simulations on tidally heated
convection in the icy shell of Europa are discussed in
Sect.~\ref{sect_eur_is}. Finally, Sect.~\ref{sect_conclusion}
concludes the paper.

\section{Ice rheology}
\label{sect_ice_rheology}

\subsection{Structure of ice}
\label{sect_ice_structure}

Water ice can exist in a great variety of different phases
(Fig.~\ref{fig_ice_phases}, Ref.~\cite{petrenko_whitworth_99}). Most
of these phases form under high pressure, which results in a
denser packing of the water molecules compared to the ``ordinary''
ice~Ih. The stability range of the latter is for pressures
$P\apprle{}200\;\mathrm{MPa}$, which is equivalent to the
hydrostatic pressure of an ice layer (density
$\rho_\mathrm{i}=910\,\mathrm{kg\,m^{-3}}$) of approximately 22~km
on Earth (gravity acceleration $g=9.81\,\mathrm{m\,s^{-2}}$),
60~km on Mars ($g=3.72\,\mathrm{m\,s^{-2}}$) and 165~km on Europa
($g=1.32\,\mathrm{m\,s^{-2}}$) and Titan
($g=1.35\,\mathrm{m\,s^{-2}}$). We will limit our discussions to
the outer shells of the planetary bodies, so that only ice~Ih
needs to be considered in the following.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.8]{Fig01_ice_phases.pdf}
  \caption{Phase diagram of the ice-water system.
  Only stable phases are shown.
  Figure by Petrenko and Whitworth
  \cite[their Fig.~11.2 on p.~253]{petrenko_whitworth_99}.}
  \label{fig_ice_phases}
\end{figure}

Ice~Ih forms hexagonal crystals, that is, the water molecules are
arranged in layers of hexagonal rings
(Fig.~\ref{fig_iceIh}, Ref.~\cite{paterson_94}).
The plane of such a layer is called the basal
plane, which actually consists of two planes shifted slightly (by
0.0923~nm) against each other. The direction perpendicular to the
basal planes is the optic axis or c-axis, and the distance between
two subsequent basal planes is 0.276~nm. This leads to the very
low packing factor of 34\%, which is responsible for the density
anomaly of ice~Ih (smaller density than liquid water). Further, the
basal planes can glide on each other when a shear stress is
applied, comparable to the deformation of a deck of cards. This
effect is strongly enhanced by the existence and generation of
dislocations (structural defects) in real crystals, and the
mechanism is consequently called \emph{dislocation creep}.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.92]{Fig02_iceIh.pdf}
  \caption{Structure of an ice crystal. The circles denote the oxygen
  atoms of the H$_2$O molecules. (a) Projection on the basal plane.
  (b) Projection on plane indicated by the broken line in (a).
  Figure by Paterson \cite[his Fig.~5.1 on p.~80]{paterson_94}.}
  \label{fig_iceIh}
\end{figure}

On the macro-scale, ice aggregates on planetary bodies (grounded
ice sheet/cap/gla\-cier, ground ice, floating ice shell etc.) are
composed of a vast number of individual crystals. For instance,
for terrestrial ice sheets and glaciers, the typical grain size is
of the order of millimeters to centimeters. Such a compound is
called \emph{polycrystalline ice}. At the time of formation, it
can be assumed that the orientation of the crystals is completely
at random, so that the macroscopic behaviour of the compound will
be isotropic. Ice-core studies on Earth have revealed that in the
course of time anisotropic fabrics can develop due to the strain
history which a piece of ice experiences during its motion in an
ice sheet or glacier (e.g.\ \cite{azuma_etal_00,
thorsteinsson_96}; see also \cite[this volume]{placidi_etal_06}).
However, since essentially nothing is known about the fabric of
extraterrestrial ice bodies, we will assume isotropic conditions
for simplicity in the following.

\subsection{Pressure melting point}
\label{sect_pressure_melting}

The melting temperature of pure ice~Ih depends on the pressure
$P$. It can therefore be written as
\begin{equation}
  T_\mathrm{m} = T_0 - f(P),
  \label{eq_p_melt_temp}
\end{equation}
where $T_0=273.16\;\mathrm{K}$ is the melting temperature for
$P_0=611.657\,\mathrm{Pa}$, that is, at the triple point of water.
According to \cite{wagner_etal_94}, the melting-point depression
$f(P)$ is in implicit form
\begin{equation}
  \begin{array}{c}
    \bigfrac{P}{P_0}
    = 1-\mbox{626,000}\,(1-\Theta^{-3})+\mbox{197,135}\,(1-\Theta^{21.2}),
    \\[2ex]
    \Theta = \bigfrac{T_\mathrm{m}}{T_0} = \bigfrac{T_0-f(P)}{T_0},
  \end{array}
  \label{eq_melt_wagner}
\end{equation}
where $P_0\le{}P\le{}209.9\,\mathrm{MPa}$ and
$T_0\ge{}T_\mathrm{m}\ge{}251.165\,\mathrm{K}$
(Fig.~\ref{fig_ice_melt}). For moderate pressures,
Eq.~(\ref{eq_melt_wagner}) can be linearized as
\begin{equation}
  f(P) = \beta{}P,
  \label{eq_melt_linear}
\end{equation}
where $\beta=7.42\times{}10^{-8}\,\mathrm{K\,Pa^{-1}}$ is the
Clausius-Clapeyron constant for pure ice \cite{paterson_94}. The
error of this linearization is negligible for $P\le{}10\,\mathrm{MPa}$,
less than 10\% for $P\le{}50\,\mathrm{MPa}$ and approximately 30\% for
$P\approx{}200\,\mathrm{MPa}$ (Fig.~\ref{fig_ice_melt}).

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.8]{Fig03_ice_melt.pdf}
  \caption{Pressure melting point of ice~Ih by
  Wagner et~al.\ \cite{wagner_etal_94}
  [Eq.~(\ref{eq_melt_wagner}), solid line] and linearized
  relation (\ref{eq_melt_linear}) (dashed line).}
  \label{fig_ice_melt}
\end{figure}

Impurities of different kinds alter the melting point further. For
terrestrial, air-saturated glacier ice, Paterson \cite{paterson_94}
reports a modified Clausius-Clapeyron constant of
$\beta=9.8\times{}10^{-8}\,\mathrm{K\,Pa^{-1}}$ for the linearized
melting-point depression (\ref{eq_melt_linear}). If salts are
present, an additional melting-point depression occurs which
depends on the type of salts and their concentration.
Intermixtures of ammonia, which may play a role in a supposed
subsurface ocean on Titan below an outer ice-Ih layer
\cite[and references therein]{sohl_etal_03}, have a similar effect.

\subsection{Flow of polycrystalline ice}
\label{sect_ice_flow}

When a specimen of polycrystalline ice is subjected to a constant
normal or shear stress, it responds with a permanent deformation,
which continues as long as the stress is applied. Typically, an
initial, instantaneous elastic deformation of the polycrystalline
aggregate is followed by a phase called primary creep during which
the strain rate decreases continuously. This behaviour is related
to the increasing geometric incompatibilities of the deforming
single crystals with different orientations. After some time, a
minimum strain rate is reached which remains constant in the
following, so that the strain increases linearly with time. This
phase is known as secondary creep. Especially in case of high
temperatures ($\apprge{}-10\degC$), at a later stage dynamic
recrystallisation (nucleation and growth of crystals which are
favourably oriented for deformation) sets in, which leads to
accelerated creep and finally a constant strain rate significantly
larger than that of the secondary creep. This is called tertiary
creep (Fig.~\ref{fig_ice_creep}, Ref.~\cite{paterson_94}).
Therefore, it can be assumed in good approximation that in
deforming ice masses secondary creep prevails for low temperatures
($\apprle{}-10\degC$), whereas tertiary creep prevails for high
temperatures ($\apprge{}-10\degC$), so that the strain rate can be
expressed as a unique function of the stress, the ice temperature
and the pressure.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.9]{Fig04_ice_creep.jpg}
  \caption{Creep response of a sample of polycrystalline ice to a
  constant stress for high (${}\!\apprge\!\!{}-10\degC$, marked by
  $T\!\uparrow$) and low (${}\!\apprle\!\!{}-10\degC$, marked by
  $T\!\downarrow$)
  temperatures.}
  \label{fig_ice_creep}
\end{figure}

On the basis of these considerations, several forms of a
non-linear viscous rheology for the flow of polycrystalline ice
for different stress, strain-rate and temperature regimes have
been proposed. They have in common to relate the strain-rate
tensor $\vecbi{D}=\mbox{sym}\,\mbox{grad}\,\vecbi{v}$ (velocity
$\vecbi{v}$) to the Cauchy stress deviator $\vecbi{t}\dev$, and
can be subsumed as
\begin{equation}
  \vecbi{D}
  = EA(T,P)\frac{\sigma^{n-1}}{d^p}\,\vecbi{t}\dev,
  \label{eq_flowlaw_general}
\end{equation}
where $\sigma=[\mbox{tr}\,(\vecbi{t}\dev)^2/2]^{1/2}$ is the
effective stress, $n$ is the stress exponent, $d$ is the grain
size and $p$ is the grain-size exponent
(e.g.\ \cite{durham_etal_97a, goldsby_kohlstedt_97, greve_mahajan_05,
paterson_94, vanderveen_99}). The flow rate factor $A(T,P)$
depends via the Arrhenius law
\begin{equation}
  A(T,P)=A_0\,e^{-(Q+PV)/RT}
  \label{eq_arrhenius_general}
\end{equation}
on the absolute temperature $T$ and the pressure $P$, where $A_0$ is the
pre-exponential constant, $Q$ is the activation energy, $V$ is the
activation volume and $R=8.314\;\mathrm{J\;mol^{-1}\,K^{-1}}$ is the
universal gas constant. The flow enhancement factor $E$ is equal to
unity for pure ice and can deviate from unity due to the softening or
hardening effect of impurities in the ice. Since polycrystalline ice is
described as a density-preserving (incompressible) medium, the pressure
is a free field (not governed by a material equation), and the full
Cauchy stress tensor $\vecbi{t}$ is related to the traceless deviator
$\vecbi{t}\dev$ by
\begin{equation}
  \vecbi{t} = -P\,\mathbf{1} + \vecbi{t}\dev,
\end{equation}
where $\mathbf{1}$ denotes the unity tensor.

Since appropriate values for the activation volume $V$ are poorly
constrained and the pressure effect is very small for typical
thicknesses of ice sheets and caps, we account for it in an
approximate way by setting $V=0$ and measuring the temperature
relative to the pressure melting point $T_\mathrm{m}$ (see
Sect.~\ref{sect_pressure_melting}) instead. To this end, the
\emph{homologous temperature}
\begin{equation}
  T' = T - T_\mathrm{m} + T_0 = T + f(P)
\end{equation}
is introduced, and the rate factor can be simplified as
\begin{equation}
  A(T')=A_0\,e^{-Q/RT'},
  \label{eq_arrhenius}
\end{equation}
which depends now exclusively on $T'$ \cite{paterson_94,
vanderveen_99}. For more details on this approach see
Appendix~\ref{sect_pressure}.

The general power law (\ref{eq_flowlaw_general}) can be inverted
as follows. For the effective strain rate
$\delta=(\mbox{tr}\,\vecbi{D}^2/2)^{1/2}$ we obtain
\begin{equation}
  \delta = EA(T')\frac{\sigma^n}{d^p},
  \label{eq_delta_sigma}
\end{equation}
or equivalently, solved for $\sigma$,
\begin{equation}
  \sigma = [EA(T')]^{-1/n} d^{p/n}
           \delta^{1/n}.
  \label{eq_sigma_delta}
\end{equation}
Inserting this in (\ref{eq_flowlaw_general}) and solving for
$\vecbi{t}\dev$ yields
\begin{eqnarray}
  \vecbi{t}\dev
  &=& [EA(T')]^{-1} d^p
  \left( [EA(T')]^{-1/n} d^{p/n} \delta^{1/n} \right)^{1-n}
  \vecbi{D}
  \nonumber\\
  &=& [EA(T')]^{-1/n} \frac{d^{p/n}}{\delta^{1-1/n}}\,\vecbi{D}.
\end{eqnarray}
By introducing the stress enhancement factor $E_\mathrm{s}=E^{-1/n}$ and the
associated rate factor $B(T')=[A(T')]^{-1/n}$, this can be written
as
\begin{equation}
  \vecbi{t}\dev
  = E_\mathrm{s}B(T') \frac{d^{p/n}}{\delta^{1-1/n}}\,\vecbi{D}.
  \label{eq_flowlaw_inv_general}
\end{equation}

For terrestrial ice, the well-established Glen's flow law [which
actually goes back to \cite{nye_57} in the general tensorial form]
uses the stress exponent $n=3$, the grain-size exponent $p=0$ and
for the temperature range $T'\le{}263\;\mathrm{K}$ the
pre-exponential constant
$A_0=3.985\times{}10^{-13}\;\mathrm{s^{-1}\,Pa^{-3}}$ and the
activation energy $Q=60\;\mathrm{kJ\;mol^{-1}}$
\cite{paterson_94}. The rheology defined by these parameters
describes the grain-size-independent flow mechanism of dislocation
creep, which prevails in terrestrial glaciers and ice sheets. The
flow enhancement factor for ice formed during glacial periods is
often set to $E=3$, interpreted as the softening influence of very
small amounts of fine dust, approximately
$1\;\mathrm{mg\;kg^{-1}}$ with particle sizes of 0.1 to
$2\,\mu\mathrm{m}$ \cite{hammer_etal_85}. This softening is
attributed to thin films of liquid water which form around the
dust particles and lubricate ice deformation. However, at the low
temperatures expected in the NPC this effect will not be present,
and direct hardening will be the dominant influence of dust (see
Sect.~\ref{sect_dust_content}).

Durham et~al.\ \cite{durham_etal_97a} propose an alternative flow law
for grain-size-independent dislocation creep, based on laboratory
creep tests at a confining pressure of 50~MPa. For the temperature
regime $T=\mbox{195--240}\;\mathrm{K}$, which corresponds
approximately to $T'=\mbox{200--245}\;\mathrm{K}$, they report the
parameters $n=4$, $p=0$,
$A_0=1.259\times{}10^{-19}\;\mathrm{s^{-1}\,Pa^{-4}}$ and
$Q=61\;\mathrm{kJ\;mol^{-1}}$.

However, for very low temperature and strain-rate conditions, as
they are expected in the polar ice caps of Mars, it is not clear
whether dislocation creep is still the predominant creep mechanism
in the polycrystalline ice aggregate. There is evidence that
other, grain-size-dependent flow mechanisms like grain-boundary
sliding become favoured instead \cite{goldsby_kohlstedt_97}.
These can be described by the parameters $n=1.8$, $p=1.4$,
$A_0=6.20\times{}10^{-14}\;\mathrm{s^{-1}\,Pa^{-1.8}\,m^{1.4}}$
and $Q=49\;\mathrm{kJ\;mol^{-1}}$ (see also \cite{nye_00}).

The relative contributions of the several flow laws (Glen, Durham,
Goldsby-Kohlstedt; the latter will be abbreviated as ``GK'' in the
following) can be estimated as follows. For simple shear in the
$x$-$z$ plane and $E=1$, Eq.~(\ref{eq_flowlaw_general}) reduces to
\begin{equation}
  \dot{\gamma} = 2A(T')\frac{\tau^n}{d^p},
  \label{eq_flowlaw_simple_shear}
\end{equation}
where $\tau=t_{xz}\dev$ is the shear stress and
$\dot{\gamma}=\partial{}v_x/\partial{}z=2D_{xz}$ is the shear rate.
Figure~\ref{fig_shear_rate} shows the shear rates resulting from
Eq.~(\ref{eq_flowlaw_simple_shear}) for $T'=200\,\mathrm{K}$ and
the stress range from 10~kPa to 100~MPa. For GK, the grain sizes
$d=1\,\mathrm{mm}$ and $10\,\mathrm{mm}$ have been assumed, based
on an estimate for the Martian north polar cap by Greve and Mahajan
\cite{greve_mahajan_05}. This range is also typical for
terrestrial ice sheets and glaciers.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.8]{Fig05_shear_rate.jpg}
  \caption{Shear rate $\dot{\gamma}$ vs.\ shear stress $\tau$
  for simple shear, computed by Eq.~(\ref{eq_flowlaw_simple_shear})
  for $T'=200\;\mathrm{K}$.
  Flow laws: Glen ($n=3$), Durham ($n=4$),
  GK ($n=1.8$, $p=1.4$, $d=1\;\mathrm{mm}$),
  GK ($n=1.8$, $p=1.4$, $d=10\;\mathrm{mm}$).}
  \label{fig_shear_rate}
\end{figure}

It becomes evident that the relative contributions of the
different flow laws vary strongly over this stress range. For low
stresses, grain-size dependent flow with a low stress exponent
(GK) dominates, whereas for higher stresses dislocation creep with
a higher stress exponent (Glen, Durham) becomes more important.
For the shown example with $T'=200\,\mathrm{K}$, the crossover
stresses are 166.3~kPa for Glen/GK$_{d=1\,\mathrm{mm}}$, 11.3~kPa
for Glen/GK$_{d=10\,\mathrm{mm}}$, 834.0~kPa for
Durham/GK$_{d=1\,\mathrm{mm}}$ and 192.7~kPa for
Durham/GK$_{d=10\,\mathrm{mm}}$.

In order to account for the contributions of grain-size-independent
dislocation creep and other, grain-size-dependent flow mechanisms
simultaneously, a modified version of the flow law proposed by Pettit
and Waddington \cite{pettit_waddington_03} may be used. It sums up the
two contributions via
\begin{equation}
  \vecbi{D}
  = \left( E_1 A_1(T') \frac{\sigma^{n_1-1}}{d^{p_1}}
         + E_2 A_2(T')\frac{\sigma^{n_2-1}}{d^{p_2}} \right)
    \,\vecbi{t}\dev,
\end{equation}
where
\begin{equation}
  A_{1}(T')=(A_0)_{1}\,e^{-Q_{1}/RT'},
  \quad
  A_{2}(T')=(A_0)_{2}\,e^{-Q_{2}/RT'}.
\end{equation}
In this representation, the index ``1'' refers to the parameters
of either Glen's or Durham's flow law, the index ``2'' to those of
the Goldsby-Kohlstedt flow law. However, this approach will not be
further pursued here.

Note that, by introducing the \emph{viscosity} $\eta$, the flow
law (\ref{eq_flowlaw_general}) with the Arrhenius law in the form
(\ref{eq_arrhenius}) can be written in compact form as
\begin{equation}
  \vecbi{D} = \frac{1}{2\eta(T',\sigma,d)}\,\vecbi{t}\dev,
  \qquad\mbox{where}\;\,
  \eta(T',\sigma,d) = \frac{1}{2EA(T')}\frac{d^p}{\sigma^{n-1}}.
  \label{eq_flowlaw_visc}
\end{equation}
Similarly, the inverse flow law (\ref{eq_flowlaw_inv_general})
reads
\begin{equation}
  \vecbi{t}\dev = 2\eta(T',\delta,d)\,\vecbi{D},
  \qquad\mbox{where}\;\,
  \eta(T',\delta,d) = \frac{E_\mathrm{s}B(T')}{2}
                      \frac{d^{p/n}}{\delta^{1-1/n}}.
  \label{eq_flowlaw_inv_visc}
\end{equation}
Of course, the two representations of the viscosity in
(\ref{eq_flowlaw_visc}) and (\ref{eq_flowlaw_inv_visc}) are
equivalent.

Evidently, for stress exponents $n>1$, the viscosities
(\ref{eq_flowlaw_visc}), (\ref{eq_flowlaw_inv_visc})
become infinite in the low-stress/deformation limit
$\sigma,\,\delta\rightarrow{}0$, and there has been a long-standing
debate whether this is physically acceptable. Also, some experimental
evidence seems to indicate a Newtonian ($n=1$) rheology for
this regime, even though these findings have been questioned (see the
discussion by \cite{paterson_94}). There are different possibilities
of introducing a finite residual viscosity $\eta_\mathrm{res}$. The
effective stress $\sigma$ and the effective strain rate $\delta$ in
the expressions (\ref{eq_flowlaw_visc}), (\ref{eq_flowlaw_inv_visc})
can be replaced by $\max(\sigma,\,\sigma_\mathrm{res})$ and
$\max(\delta,\,\delta_\mathrm{res})$, respectively, where
$\sigma_\mathrm{res}$ and $\delta_\mathrm{res}$ are small positive
parameters. Alternatively, a small residual viscosity may be directly
added,
\begin{equation}
  \eta(T',\sigma,d) = \frac{1}{2EA(T')}\frac{d^p}{\sigma^{n-1}}
                      + \eta_\mathrm{res},
\end{equation}
or
\begin{equation}
  \eta(T',\delta,d) = \frac{E_\mathrm{s}B(T')}{2}
                      \frac{d^{p/n}}{\delta^{1-1/n}}
                      + \eta_\mathrm{res}.
\end{equation}
The disadvantage of these forms is that they are not equivalent
anymore, and an inversion according to
Eqs.~(\ref{eq_delta_sigma})-(\ref{eq_flowlaw_inv_general}) can no
longer be executed.

\subsection{Influence of liquid water}
\label{sect_water_content}

If the temperature of ice reaches the pressure melting point,
liquid water may be present within the ice matrix, which is stored
as lenses at the grain boundaries and in capillary channels at
triple grain junctions. Let $\varphi_\mathrm{w}$ be the volume
fraction of water, then the density of the ice-water mixture is
\begin{equation}
  \rho = (1-\varphi_\mathrm{w}) \rho_\mathrm{i}
         + \varphi_\mathrm{w}\rho_\mathrm{w},
  \label{eq_rho_temp_ice}
\end{equation}
where $\rho_\mathrm{i}=910\,\mathrm{kg\,m^{-3}}$ is the density of
pure ice and $\rho_\mathrm{w}=1000\,\mathrm{kg\,m^{-3}}$ the
density of pure water. Further, it is clear that the presence of
liquid water will reduce the viscosity.
Tobie et~al.\ \cite{tobie_etal_03}
propose an exponential softening, which can be expressed by a
stress enhancement factor $E_\mathrm{s}<1$,
\begin{equation}
  E_\mathrm{s} = e^{-b_\mathrm{w}\varphi_\mathrm{w}},
  \label{eq_es_temp_ice}
\end{equation}
or, equivalently, by a flow enhancement factor $E>1$,
\begin{equation}
  E = E_\mathrm{s}^{-n} = e^{nb_\mathrm{w}\varphi_\mathrm{w}},
  \label{eq_e_temp_ice}
\end{equation}
where $b_\mathrm{w}=45$. This value was chosen such that 5\% melt
decreases the viscosity in Eq.~(\ref{eq_flowlaw_inv_visc}) by an
order of magnitude, that is,
$E_\mathrm{s}(\varphi_\mathrm{w}\!=\!0.05)=0.1$. In terrestrial
glaciology, a linear relation reported by Paterson \cite{paterson_94}
has been widely used instead, which reads in terms of the enhancement
factors
\begin{equation}
  E = 1 + b_\mathrm{w}\varphi_\mathrm{w}
  \quad\Leftrightarrow\quad
  E_\mathrm{s}
  = E^{-1/n}
  = (1 + b_\mathrm{w}\varphi_\mathrm{w})^{-1/n},
  \label{eq_e_lin_temp_ice}
\end{equation}
where $b_\mathrm{w}=\frac{580}{3.2}=181.25$. This relation has
been established based on laboratory measurements for small water
contents below 1\% ($\varphi_\mathrm{w}<0.01$).

\subsection{Influence of dust}
\label{sect_dust_content}

Satellite imagery shows that parts of the Martian polar ice caps
appear dark, which indicates that they consist of ice with some amount
of mixed-in dust. Greve and Mahajan \cite{greve_mahajan_05} have laid
down that this affects the ice flow in a multiple way, in that direct
hardening can be partly compensated or even overcompensated by the
increasing density, which increases the driving stresses, and the
decreasing heat conductivity, which makes the ice at depth warmer and
therefore softer. Therefore, the average volume fraction
$\varphi_\mathrm{d}$ of dust is introduced, and the density, $\rho$,
heat conductivity, $\kappa$, and specific heat, $c$, of the ice-dust
mixture are computed as volume-fraction-weighed averages of the values
for pure ice and crustal material,
\begin{eqnarray}
  \rho &=& (1-\varphi_\mathrm{d}) \rho_\mathrm{i}
           + \varphi_\mathrm{d} \rho_\mathrm{c},
  \nonumber\\
  \kappa &=& (1-\varphi_\mathrm{d}) \kappa_\mathrm{i}
             + \varphi_\mathrm{d} \kappa_\mathrm{c},
  \label{eq_rho_kappa_c_mixture}
  \\
  \rho c &=& (1-\varphi_\mathrm{d}) \rho_\mathrm{i} c_\mathrm{i}
             + \varphi_\mathrm{d} \rho_\mathrm{c} c_\mathrm{c},
  \nonumber
\end{eqnarray}
with the following parameters: ice density
$\rho_\mathrm{i}=910\,\mathrm{kg\,m^{-3}}$, heat conductivity of
ice
$\kappa_\mathrm{i}=9.828\,e^{-0.0057\,T[\mathrm{K}]}\,\mathrm{W\,m^{-1}K^{-1}}$,
specific heat of ice
$c_\mathrm{i}=(146.3+7.253\,T[\mathrm{K}])\,\mathrm{J\,kg^{-1}K^{-1}}$,
density of crustal material (dust)
$\rho_\mathrm{c}=2900\;\mathrm{kg\,m^{-3}}$, heat conductivity of
crustal material (dust)
$\kappa_\mathrm{c}=2.5\;\mathrm{W\,m^{-1}\,K^{-1}}$, specific heat
of crustal material (dust)
$c_\mathrm{c}=1000\,\mathrm{J\,kg^{-1}K^{-1}}$. The additional
density factors in Eq.~(\ref{eq_rho_kappa_c_mixture})$_3$ are
necessary because the averaging procedure requires volumetric
quantities, whereas the specific heat is taken per mass unit.

Direct hardening is described by a stress enhancement factor
$E_\mathrm{s}>1$ based on laboratory measurements of the
deformation of ice-dust compounds,
\begin{equation}
  E_\mathrm{s} = e^{b_\mathrm{d}\varphi_\mathrm{d}},
\end{equation}
where $b_\mathrm{d}=2$ and $\varphi_\mathrm{d}\le{}0.56$
\cite{durham_etal_97a}. This is equivalent to a flow enhancement
factor
\begin{equation}
  E = E_\mathrm{s}^{-n} = e^{-nb_\mathrm{d}\varphi_\mathrm{d}}.
  \label{eq_enh_mixture}
\end{equation}
Hence, for given stress, temperature and grain-size conditions and
a stress exponent $n=3$ a dust content of 10\%
($\varphi_\mathrm{d}=0.1$) leads to an almost twice as hard
material ($E=0.55$) compared to pure ice.

\section{Model equations for ice flow}
\label{sect_model_ice_flow}

The thermo-mechanical problem of ice flow in a planetary
environment can be described by the balance equations of mass,
momentum and energy (e.g.\ \cite{hutter_83}). For a
density-preserving (incompressible) medium, which holds for ice in
good approximation despite the variability expressed by
Eqs.~(\ref{eq_rho_temp_ice}) and
(\ref{eq_rho_kappa_c_mixture})$_1$, the mass balance (continuity
equation) reads
\begin{equation}
   \mbox{div}\,\vecbi{v} = 0.
  \label{eq_mass_balance}
\end{equation}
For evolving ice bodies, it is convenient to vertically integrate
the continuity equation. This yields the ice-thickness ($H$)
equation
\begin{equation}
  \pabl{H}{t} = -\mbox{div}\,\vecbi{Q} + a_\mathrm{s} - a_\mathrm{b},
  \label{eq_ice_thickness}
\end{equation}
where $\vecbi{Q}$ is the volume flux (vertically integrated
horizontal velocity), and $a_\mathrm{s}$ and $a_\mathrm{b}$ are
the mass balances at the surface (positive for supply) and the
bottom (positive for loss), respectively. The momentum balance
yields with the flow law (\ref{eq_flowlaw_inv_visc}) the
Stokes equation
\begin{equation}
  - \mbox{grad}\, P
  + \mbox{div}\,\!\!\left[\eta\left(\mbox{grad}\,\vecbi{v}+(\mbox{grad}\,\vecbi{v})\tra\right)\right]
  + \rho\vecbi{g} = \mathbf{0}
  \label{eq_stokes}
\end{equation}
($\vecbi{g}$: vectorial gravity acceleration), in which the
acceleration term $\rho\,\mathrm{d}\vecbi{v}/\mathrm{d}{}t$ has been neglected due
to the very low flow velocities to be expected. From the energy
balance, Fourier's law of heat conduction
\begin{equation}
  \vecbi{q} = -\kappa(T)\,\mbox{grad}\,{}T
  \label{eq_fourier_heat_conduction}
\end{equation}
($\vecbi{q}$: heat flux, $\kappa$: heat conductivity) and the
caloric equation of state
\begin{equation}
  u = \int\limits_{T_0}^{T}c(\bar{T})\,\mathrm{d}\bar{T}
  \label{eq_int_energy_temp}
\end{equation}
($u$: specific internal energy, $c$: specific heat), the
temperature-evolution equation
\begin{equation}
  \rho c \left( \pabl{T}{t} + \vecbi{v}\cdot\mbox{grad}\,{}T \right)
  = \mbox{div}\,(\kappa\,\mbox{grad}\,{}T) + 4\eta\,\delta^2 + r
  \label{eq_ice_flow_temperature}
\end{equation}
results. In this relation, the production term $4\eta\,\delta^2$
is the strain heating, and the source term $r$ denotes the
volumetric heating due to radiation and tidal dissipation.

The above equations need to be complemented by dynamic and
thermodynamic boundary conditions at the surface and the bottom of
the respective ice body. If we assume that the surface is in
contact with the atmosphere, then it can be described in good
approximation as stress-free, that is,
\begin{equation}
  \left.\vecbi{t}\cdot\vecbi{n}\right|_\mathrm{s} = \mathbf{0}
\end{equation}
(where $\vecbi{n}$ is the outer normal unit vector, and the subscript
``s'' denotes the surface). The surface temperature $T_\mathrm{s}$
can be prescribed directly as a Dirichlet condition.

If the bottom is a rigid ice/rock, ice/regolith or ice/sediment
interface, no-slip conditions can be employed,
\begin{equation}
  \vecbi{v}_\mathrm{b} = \mathbf{0}
\end{equation}
(the subscript ``b'' stands for the bottom). As for the temperature
field, let us assume that the basal heat flux into the ice,
$\vecbi{q}_\mathrm{b}$, is known. This yields the Neumann condition
\begin{equation}
  \left.\kappa\pabl{T}{\vecbi{n}}\right|_\mathrm{b} = \vecbi{q}_\mathrm{b},
\end{equation}
where $\vecbi{n}$ is again the outer normal unit vector.

The situation is different if the bottom is an ice/water interface.
In this case, the basal stress conditions are governed by the
hydrostatic pressure $P_\mathrm{b}$ of the water at the interface,
\begin{equation}
  \left.\vecbi{t}\cdot\vecbi{n}\right|_\mathrm{b}
  = -P_\mathrm{b}\,\vecbi{n},
\end{equation}
and the bottom temperature equals the pressure melting point,
\begin{equation}
   T_\mathrm{b} = T_\mathrm{m}.
\end{equation}
Provided that the role of impurities is negligible, $T_\mathrm{m}$ can
be obtained from Eq.~(\ref{eq_p_melt_temp}).

\section{Polar caps of Mars}
\label{sect_mpc}

\subsection{Large-scale simulations}
\label{sect_mpc_sico}

The dynamic and thermodynamic state of the present-day polar caps
of Mars will now be simulated with the ice-sheet model SICOPOLIS
(``SImulation COde for POLythermal Ice Sheets''). This model was
developed in the mid-1990's for terrestrial applications and has
later been adapted to the north-polar cap of Mars (see
\cite{greve_mahajan_05}, and references therein).  It solves the
ice-flow equations described in Sect.~\ref{sect_model_ice_flow}
based on the shallow-ice approximation \cite{hutter_83,
morland_84}, that is, the flow regime is assumed to be simple,
bed-parallel shear, the pressure is hydrostatic and lateral shear
stresses as well as normal stress deviators are neglected (see
also \cite[this volume]{calov_06}). Inputs from the environment
are specified by the mean annual surface temperature, the net
surface mass balance (ice accumulation minus ablation) and the
basal (``geothermal'') heat flux from the underlying lithosphere.
The numerical solution of the model equations is carried out by a
finite-difference integration technique.

Here, we only consider steady-state conditions, that is, the ice
surface is held fixed as given by the Mars Orbiter Laser Altimeter
(MOLA) data of the Mars Global Surveyor (MGS) spacecraft
\cite{smith_etal_99a, zuber_etal_98}. This makes it unnecessary
to prescribe the surface mass balance. The topographies are shown
in Figs.~\ref{fig_nmars} and \ref{fig_smars} (top left panels) for
the north- and south-polar caps and their surroundings. It is
striking that the overall shape of the north-polar cap is quite
regular and smooth, whereas the south-polar cap appears much
more rugged.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=1.2]{Fig06_nmars.pdf}
  \caption{North-polar cap of Mars:
  MOLA surface topography $h$ \cite{smith_etal_99a},
  computed ice thickness $H$, computed surface velocity $v_\mathrm{s}$
  and computed homologous basal temperature $T'_\mathrm{b}$ for
  present-day steady-state conditions. Heavy-dashed lines indicate the
  ice-cap margin.}
  \label{fig_nmars}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[scale=1.2]{Fig07_smars.pdf}
  \caption{South-polar cap of Mars:
  MOLA surface topography $h$ \cite{smith_etal_99a},
  computed ice thickness $H$, computed surface velocity $v_\mathrm{s}$
  and computed homologous basal temperature $T'_\mathrm{b}$ for
  present-day steady-state conditions. Heavy-dashed lines indicate the
  ice-cap margin.}
  \label{fig_smars}
\end{figure}

The ice temperature, velocity and bottom topography are allowed to
evolve freely until the steady state is reached. Surface
temperature is delivered by the model by Grieger (pers.\ comm.\
2004; see also http://www.space-vision.biz/marstemperatures.html),
which is based on a zonal and daily mean energy balance including
a simple scheme for CO$_2$ condensation and evaporation. Results
agree well with those of the Martian Climate Database
\cite{lewis_etal_99}; however, both methods fail in reproducing
the observed year-round CO$_2$ cover of the southern residual cap.
Therefore, the southern mean-annual surface temperature is
corrected such that it equals the CO$_2$ sublimation temperature
of $-128\degC$ within $85\degS$. Further, the geothermal heat flux
is assumed to be $35\,\mathrm{mW\,m^{-2}}$
\cite{schubert_etal_92}, and the source term $r$ in the
temperature evolution equation (\ref{eq_ice_flow_temperature}) is
neglected.

The unknown topography of the solid ground below the polar caps is
computed in two steps. First, the equilibrated ground for ice-free
conditions, which is required as a reference topography, is determined
by a smooth extrapolation of the ice-free ground surrounding the north
and south polar cap, respectively \cite{greve_etal_04a}. Second, the
actual ground topography is obtained by superposing the isostatic
deflection (downward displacement $w$) of the underlying lithosphere
due to the ice load. By modelling the lithosphere as a thin elastic
plate, the isostatic deflection is governed by the bi-potential
equation
\begin{equation}
  K_\mathrm{l}\,\nabla^4{}w = \rho g H - \rho_\mathrm{a} g w,
  \label{eq_elast_litho}
\end{equation}
where $K_\mathrm{l}=10^{25}\,\mathrm{N\,m}$ is the flexural stiffness
of the lithosphere, $\nabla^4$ the bi-potential operator in the
horizontal plane and $\rho_\mathrm{a}=3500\,\mathrm{kg\,m^{-3}}$ the
density of the asthenosphere (viscous mantle layer below the elastic
lithosphere). The two load terms on the right-hand side are the ice
load itself ($\rho{}gH$) and the counteracting buoyancy force which
the deflected lithosphere experiences from the asthenosphere below
($\rho_\mathrm{a}gw$). Note that the ice thickness $H$ depends on $w$
and is therefore part of the solution. In transient scenarios, which
are not considered here, the elastic deflection (\ref{eq_elast_litho})
is not assumed instantaneously, so that an additional evolution
equation is required for the non-equilibrium displacement
\cite{greve_01, lemeur_huy_96}.

The simulations have been carried out by applying Glen's flow law
as discussed in Sect.~\ref{sect_ice_flow}. The dust content is
assumed to be 20\% ($\varphi_\mathrm{d}=0.2$, see
Sect.~\ref{sect_dust_content}). Horizontal resolution is 20~km in
the stereographic plane with standard parallel $71\degN$/S,
vertical resolution is 51 grid points in the cold-ice column and
11 grid points in the lithosphere column, and the time-step is
1000 years.

Results for the ice thickness, the surface velocity and the
homologous basal temperature are shown in Figs.~\ref{fig_nmars}
and \ref{fig_smars} (top right and bottom panels) for both polar
caps. The north-polar cap assumes its maximum thickness of 3.62~km
almost exactly at the pole, whereas the maximum thickness of the
south-polar cap, 3.88~km, is offset by approx.\ 150~km in
$10\degW$ direction. Of the entire ice volume of the NPC,
which is $1.53\times{}10^6\,\mathrm{km}^3$, a fraction of
$1.23\times{}10^6\,\mathrm{km}^3$ (80\%) is above the level of the
equilibrated ground, and the remaining 20\% are due to the
isostatic deflection of the lithosphere. This ratio is almost the
same for the SPC, with a total volume of
$2.30\times{}10^6\,\mathrm{km}^3$ and a volume above the
equilibrated ground of $1.85\times{}10^6\,\mathrm{km}^3$.

Compared to terrestrial ice sheets and glaciers with typical flow
velocities of tens to hundreds of meters per year, the flow of the
Martian polar caps is very slow. The NPC reaches a maximum surface
velocity of $0.98\,\mathrm{mm\,a^{-1}}$, and south of $85\degN$
surface velocities are everywhere less than
$0.01\,\mathrm{mm\,a^{-1}}$. Therefore, the active, dynamic zone
only consists of the interior, thick part of the ice cap. The
situation is even more extreme for the SPC, where the maximum
surface velocity is as small as $0.055\,\mathrm{mm\,a^{-1}}$, and
by far the largest part of the ice cap flows at speeds slower than
a micrometer per year (note the different scales of the colour
bars). Therefore, the SPC can be considered as essentially
stagnant.

Note that the 20-km resolution applied here does not resolve
small-scale structures like the scarp/trough systems, which may
lead to locally enhanced flow velocities. This will be discussed
below (Sect.~\ref{sect_mpc_scarp}) for the NPC.

The slow flow velocities of the Martian polar caps lead to a
conduction-dominated heat transport within the ice. The ratio
of convection to conduction is given by the \emph{Peclet number}
\begin{equation}
  P\!e = \frac{[U][L]}{[\alpha]},
\end{equation}
where $[U]$ is a velocity scale, $[L]$ is a scale of extent and
$[\alpha]$ is a scale for the thermal diffusivity
$\alpha=\kappa/(\rho{}c)$. With $[U]=0.1\,\mathrm{mm\,a^{-1}}$ for
the NPC and $0.01\,\mathrm{mm\,a^{-1}}$ for the SPC, respectively,
$[L]=300\,\mathrm{km}$ (only the inner, dynamic region considered;
see velocity panels of Figs.~\ref{fig_nmars} and \ref{fig_smars})
and $[\alpha]=2\times{}10^{-6}\,\mathrm{m^2\,s^{-1}}$ (for
$-100\degC$ and 20\% dust), one obtains Peclet numbers of
$P\!e\approx{}0.5$ for the NPC and $P\!e\approx{}0.05$ for the
SPC, respectively. Evidently, both values are less than unity, so
that conduction outweighs convection. This leads to temperature
profiles which increase essentially linearly with depth (not
shown). By contrast, for terrestrial ice sheets and ice caps,
$P\!e\gg{}1$ always holds, so that their temperature fields are
mainly governed by heat convection.

The different flow behaviour of the NPC and SPC is mainly due to
the lower ice temperatures of the SPC, which are a consequence of
the $\sim{}20\degC$ difference in surface temperatures. Since the
thicknesses of both caps are similar, and the temperature
distribution is mainly controlled by heat conduction, this
transfers directly to a $\sim{}20\degC$ difference in average
basal temperatures (again, note the different scales of the colour
bars in the respective panels of Figs.~\ref{fig_nmars} and
\ref{fig_smars}). The highest homologous basal temperatures are
$-69.1\degC$ for the NPC and $-89.7\degC$ for the SPC,
respectively, which demonstrates the insulating effect of the ice
caps against the much colder surface temperatures. Nevertheless,
the basal temperatures are in any case far below the pressure
melting point. This is a very robust result, so that the presence
of large amounts of subglacial liquid water as a potential habitat
for Martian lifeforms can essentially be ruled out.

The simulation for the north-polar cap has been re-run with the
alternative flow laws shown in Fig.~\ref{fig_shear_rate} (GK with
$d=1\,\mathrm{mm}$ and $10\,\mathrm{mm}$, Durham). As it was already
found by Greve and Mahajan \cite{greve_mahajan_05}, this has a very
significant influence on the computed flow velocities. The maximum
surface velocity varies by almost three orders of magnitude, from the
largest value $3.02\,\mathrm{mm\,a^{-1}}$ for GK$_{d=1\,\mathrm{mm}}$
via the above-mentioned $0.98\,\mathrm{mm\,a^{-1}}$ for the reference
simulation (Glen) and $0.12\,\mathrm{mm\,a^{-1}}$ for
GK$_{d=10\,\mathrm{mm}}$ to only $0.013\,\mathrm{mm\,a^{-1}}$ for
Durham. As expected, this order corresponds to the low-stress regime
in Fig.~\ref{fig_shear_rate}. Since the creep mechanism is probably a
combination of dislocation creep (Glen, Durham) and
grain-size-dependent creep (GK), we consider a value of the order of
$1\,\mathrm{mm\,a^{-1}}$ as most likely. By contrast, the basal
temperature is virtually unaffected by the assumed flow law, which is
a consequence of the dominance of flow-independent heat conduction
over flow-dependent heat convection.

\subsection{Detailed simulations of the scarps and troughs}
\label{sect_mpc_scarp}

In the large-scale simulations discussed above
(Sect.~\ref{sect_mpc_sico}), the spiralling scarps and troughs
which cut up to several hundred meters into the surface of the
Martian polar caps, have not been resolved. This was tackled in
a study by Hvidberg \cite{hvidberg_03}, where a flowline of the
north-polar cap extending from the pole in $160\degE$ direction
was investigated in detail, and an axisymmetric ice cap was
assumed. The flowline was discretized by a finite-element grid at
kilometer-scale horizontal resolution, and the ice-flow equations
of Sect.~\ref{sect_model_ice_flow} were solved without further
approximations. Otherwise, the model set-up is very similar to
that described in Sect.~\ref{sect_mpc_sico}.

The temperature and flow fields computed by Hvidberg
\cite{hvidberg_03} are shown in Fig.~\ref{fig_hvidberg_fig2}. While
the overall values agree very well with those reported above for the
large-scale simulations (Sect.~\ref{sect_mpc_sico},
Fig.~\ref{fig_nmars}), it becomes evident that the topographic
disturbances imposed by the scarps/troughs result in significant
disturbances of the ice flow which propagate all the way down to the
bottom. The local flow accelerations are highlighted in
Fig.~\ref{fig_hvidberg_fig3}, which shows the surface velocities and
the surface mass balance required for maintaining the steady state.
For the most pronounced trough about 220~km away from the pole, the
horizontal surface velocity is larger than $15\,\mathrm{mm\,a^{-1}}$,
and even the vertical velocity reaches values of almost
$\pm{}10\,\mathrm{mm\,a^{-1}}$, distributed such that the trough would
close in the absence of any mass exchange.  In order to keep the
troughs open, this must be balanced by a mass exchange at the surface
such that ice accumulates outside the troughs, but is removed from
within them at a rate of some millimeters per year
(Fig.~\ref{fig_hvidberg_fig3}, bottom panel).  This exchange pattern
was already proposed by Fisher \cite{fisher_93, fisher_00} and termed
``accublation model''.  Physical processes behind the ``accublation''
exchange are likely differential ablation due to the albedo contrast
(white material outside vs.\ darker material inside the troughs)
and/or enhanced wind erosion due to the formation of local turbulences
in the troughs.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.94]{Fig08_hvidberg_fig2.pdf}
  \caption{Computed temperature and velocity field of the north
  polar cap of Mars, for a flowline extending southward from the pole in
  $160\degE$ direction.
  Figure by Hvidberg \cite[her Fig.~2]{hvidberg_03}.}
  \label{fig_hvidberg_fig2}
\end{figure}

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.94]{Fig09_hvidberg_fig3.pdf}
  \caption{MOLA surface topography, computed surface velocity and computed
  steady-state mass balance corresponding to the simulation shown in
  Fig.~\ref{fig_hvidberg_fig2}.
  Figure by Hvidberg \cite[her Fig.~3]{hvidberg_03}.}
  \label{fig_hvidberg_fig3}
\end{figure}

\clearpage

\section{Icy shell of Europa}
\label{sect_eur_is}

While the polar caps of Mars are localized ice masses on the
planet's surface which rest on solid land, the icy shell of Europa
envelops the entire planetary body and is underlain most likely by
a deep ocean. Also, the supposed thickness within the range of
10--50~km is much larger than that of the Martian polar caps. This
allows the formation of convection cells, so that vertical motion
may play a much larger role.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.95]{Fig10_tobie_etal_fig1.pdf}
  \caption{Model of the radial structure of Europa with a 20-km
  thick outer ice shell underlain by a deep ocean. The ice shell
  consists of an upper conductive lid and a lower convective
  layer. A possible temperature profile is shown in the inset plot.
  Figure by Tobie et~al.\ \cite[their Fig.~1]{tobie_etal_03}.}
  \label{fig_tobie_etal_fig1}
\end{figure}

Tobie et~al.\ \cite{tobie_etal_03} investigated this problem by
assuming a 20-km thick ice shell with a surface temperature of 100~K
and a bottom temperature of 270~K, which is approximately equal to the
pressure melting point (Fig.~\ref{fig_tobie_etal_fig1}). The ice-flow
equations of Sect.~\ref{sect_model_ice_flow} are solved by finite
differences in a two-dimensional cross-section of 40~km width, and the
domain is treated as Cartesian. For the viscosity law, a simplified
Newtonian viscosity is assumed, that is, $n=1$ and $p=0$ [see
Eqs.~(\ref{eq_flowlaw_visc}) and (\ref{eq_flowlaw_inv_visc})], and the
dependence on the homologous temperature $T'$ is modelled by an
activation energy of $Q=50\;\mathrm{kJ\;mol^{-1}}$, which is close to
the value of Goldsby and Kohlstedt \cite{goldsby_kohlstedt_97}. The
bottom viscosity is set to $1.5\times{}10^{14}\,\mathrm{Pa\,s}$ in the
reference simulation.  The tidal dissipation $r$ in the energy balance
(\ref{eq_ice_flow_temperature}) is calculated by assuming a
viscoelastic response of Europa to the tidal forcing, and the
parameters of the Maxwellian rheology are prescribed variably in the
four layers indicated in Fig.~\ref{fig_tobie_etal_fig1}. This can lead
to partial melting in the lower parts of the ice shell, the effect of
which is accounted for by
Eqs.~(\ref{eq_rho_temp_ice})--(\ref{eq_e_temp_ice}). In the vertical
component of the Stokes equation (\ref{eq_stokes}), the buoyancy force
due the varying ice density, which results from partial melting and
thermal expansion, is added.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=0.95]{Fig11_tobie_etal_fig4.pdf}
  \caption{Computed flow field (arrows in both panels),
  temperature (panel a) and tidal heating (panel b) in
  Europa's ice shell for the reference simulation by
  Tobie et~al.\ \cite[their Fig.~4]{tobie_etal_03}.
  The solid contour in panel (a) delimits the area where the temperature
  is at the pressure melting point.}
  \label{fig_tobie_etal_fig4}
\end{figure}

The resulting fields of ice flow, temperature and tidal heating are
shown in Fig.~\ref{fig_tobie_etal_fig4}. As already sketched in
Fig.~\ref{fig_tobie_etal_fig1}, the huge temperature (and therefore
viscosity) differences lead to the formation of an upper, conductive
sublayer of approximately 8~km thickness, which is essentially rigid,
and in which the temperature increases linearly with depth. By
contrast, in the underlying convective sublayer significant ice flow
occurs in the form of convection cells, with a maximum horizontal
velocity of $0.29\,\mathrm{m\,a^{-1}}$, a maximum upward velocity of
$0.265\,\mathrm{m\,a^{-1}}$ and a maximum downward velocity of
$0.352\,\mathrm{m\,a^{-1}}$. Therefore, vertical temperature
gradients are small in the convective sublayer. In the rising plumes,
the ice temperature reaches the pressure melting point, and
consequently partial melting occurs. The tidal heating is negligible
in the upper, conductive sublayer, but reaches significant values up
to $2.7\times{}10^{-6}\,\mathrm{W\,m^{-3}}$ in the lower parts of the
ice shell. It is also interesting to note that the convection cells
entail large lateral gradients of the temperature and the tidal
heating.

Fig.~\ref{fig_tobie_etal_fig6} shows the corresponding heat flux
at the surface and the bottom of the ice shell (solid line in the
plots). As it can be expected, the lateral variability of the heat
flux at the surface is rather small, and the average value is
approximately $40\,\mathrm{mW\,m^{-2}}$. By contrast, at the
bottom, the convection cells produce a heat-flux pattern highly
variable in space and time (the latter is not shown in the
snapshot figure) with an average of about one fourth of the
surface heat flux.

\begin{figure}[htb]
  \centering
  \includegraphics[scale=1.0]{Fig12_tobie_etal_fig6.pdf}
  \caption{Surface and bottom heat flux corresponding to the
  simulation shown in Fig.~\ref{fig_tobie_etal_fig4} (solid line).
  The dashed line is for the same simulation without tidal
  heating \cite[their Fig.~6]{tobie_etal_03}.}
  \label{fig_tobie_etal_fig6}
\end{figure}

In addition to the reference simulation discussed above, Tobie et~al.\
\cite{tobie_etal_03} studied also the effects of different bottom
viscosities, tidal dissipation rates (dashed line in
Fig.~\ref{fig_tobie_etal_fig6}, etc.) and ice-shell thicknesses.  This
shall not be further reported here.

\section{Concluding remarks}
\label{sect_conclusion}

We have seen that water ice plays an important role in the climate
systems of Earth and Mars. Further, it is a major constituent of the
bulk volumes of the satellites of Jupiter, Saturn, Uranus and Neptun
as well as the planet Pluto. The ice sheets, ice caps and glaciers on
Earth and Mars show glacial flow, driven by their own weight. By
contrast, ice flow driven by thermal convection can occur in the crust
and interior of the icy satellites, depending on the temperature
gradient between the warm interior and the cold surface. Here, we have
limited the discussions to the ``ordinary'' ice~Ih. However, at
greater depths the occurrence of high-pressure phases can be expected,
the rheology of which is poorly known.

Further ``ices'' made up of other moderately volatile substances are
known or supposed to occur, for example, on Mars (CO$_2$), Io
(SO$_2$), Europa, Ganymede and Callisto (NH$_3$), Titan (CH$_4$ and
other carbohydrates), Triton and Pluto (N$_2$, CH$_4$).  However, from
a fluid-dynamical point of view, very little is known about their
relevance, and it has not been possible to assess so far whether these
materials are significant parts of flowing ice masses somewhere in the
solar system.

\section*{Acknowledgements}

The author wishes to thank K.~Hutter for his kind invitation to
contribute this article to the current issue of the
\emph{GAMM-Mitteilungen}. Comments by R.~Calov and K.~Hutter
on the draft version of the manuscript are gratefully
acknowledged.
Figure~\ref{fig_ice_phases} reproduced by permission of Oxford University
Press.
Figure~\ref{fig_iceIh} reprinted with permission from Elsevier.
Figures~\ref{fig_hvidberg_fig2} and \ref{fig_hvidberg_fig3} reprinted
from the \emph{Annals of Glaciology} with
permission of the International Glaciological Society.
Figures~\ref{fig_tobie_etal_fig1}, \ref{fig_tobie_etal_fig4} and
\ref{fig_tobie_etal_fig6} reproduced by permission of the American
Geophysical Union.

\input{Greve_2006_GAMM-Mitt.bbl}

\begin{appendix}

\section{Pressure dependence of ice flow}
\label{sect_pressure}

The flow rate factor (\ref{eq_arrhenius}) can be written as
\begin{eqnarray}
  A(T')
  &=& A_0\exp\left(-\frac{Q}{RT'}\right)
  \nonumber\\
  &=& A_0\exp\left(-\frac{Q}{R(T+\beta{}P)}\right)
   =  A_0\exp\left(-\frac{Q}{RT(1+\beta{}P/T)}\right),
\end{eqnarray}
where the linearized melting-point depression
(\ref{eq_melt_linear}) has been used for simplicity. For the
stability range of ice~Ih, $P\apprle{}200\;\mathrm{MPa}$, we have
$\beta{}P<{}20\;\mathrm{K}$, and therefore $\beta{}P/T\ll{}1$.
This allows the Taylor approximation
\begin{equation}
  A(T')
  = A_0\exp\left(-\frac{Q}{RT}\Big(1-\frac{\beta{}P}{T}\Big)\right)
  = A_0\exp\left(-\frac{1}{RT}\Big(Q-\frac{Q\beta{}P}{T}\Big)\right).
\end{equation}
By comparing this with the flow rate factor in the general
form (\ref{eq_arrhenius_general}), the activation volume which
corresponds to the simplified form (\ref{eq_arrhenius}) is
\begin{equation}
  PV = -\frac{Q\beta{}P}{T}
  \quad\Rightarrow\quad
  V = -\frac{Q\beta{}}{T}.
\end{equation}
For $Q=60\;\mathrm{kJ\;mol^{-1}}$ and $T=200\;\mathrm{K}$,
this yields $V=-2.94\times{}10^{-5}\;\mathrm{m^3\,mol^{-1}}$,
which agrees well with the (quite uncertain) values reported by
Paterson \cite{paterson_94},
$V=-1.7\times{}10^{-5}\;\mathrm{m^3\,mol^{-1}}$, and by
Durham et~al.\ \cite{durham_etal_97a},
$V=-1.3\times{}10^{-5}\;\mathrm{m^3\,mol^{-1}}$. Therefore, the
pressure dependence of the flow rate factor is described
reasonably well by the simplified form (\ref{eq_arrhenius}) which
depends on the homologous temperature only.

\end{appendix}

\end{document}

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Episode Ignis - Coming in December
October 31st, 2017 1
Fans of Ignis from Final Fantasy XV have been waiting for what seems like forever for his story arc, and it is finally on its way! Through a video posted on Facebook today, the release date for Final Fantasy XV: Episode Ignis was revealed, as well as some footage of what the episode will...
Prompto is now playable in A New Empire
Ever since Final Fantasy XV: A New Empire was released, players have been calling for Prompto to be a playable hero. Well, the wait is finally over! The blonde paparazzo became available for use less than a week ago, and now we are able to see what advantages there are to using the photog...
Final Fantasy XV: Comrades - TGS Trailer 2017
September 22nd, 2017 0
We previously mentioned that Final Fantasy XV: Comrades will be the game's first online multiplayer expansion on October 31 2017. The setting takes place after the disappearance of Noctis and players take the role of a survivor from the Kingsglaive. As members of the Kingsglaive, players are...
Final Fantasy XV: Assassin's Festival (Assassin's Creed Collaboration Event)
September 2nd, 2017 0
Square Enix announced their collaboration between Final Fantasy XV and Ubisoft Entertainment's series Assassin's Creed which will be available for free via the Xbox Game Store and PlayStation Store. Final Fantasy XV: Assassin's Festival event will run from September 1, 2017 through January 31,...


Spare, bilingual text written in Cree (in both the standard Roman orthography and in syllabics) and in English presents a multigenerational harvesting of wild yarrow.
The first-person narration tells readers that a little girl and her mother await Nôhkom (Cree for "my grandmother") as she prepares to harvest yarrow flowers and leaves. Accompanying illustrations rendered in acrylics on canvas depict the girl and both women and then follow them as they venture out into the fields, accompanied by a brown-and-white dog that's unmentioned in the text. The palette is gentle and the style soft, with art that invites contemplative engagement in this small moment of family togetherness. In three separate full-page illustrations, each worker prays, then the work begins. "Nôhkom picks. / I pick." But—"Mom?" The book catches Mom in a playful moment, blowing at a yarrow puff. Daughter and grandmother wait for her. When the narrator pronounces the task "done!" her face is utterly overtaken by delight. Although the book ends with pictures of the girl and women with the flowers in the fields, backmatter provides a simple recipe for yarrow tea, often used medicinally, which suggests the reason for their harvest.


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Caverion modernising the telematic system on motorway E18 between Turku and Muurla in Finland
Caverion Corporation Investor news 9 June, 2017 at 10.00 a.m. EEST
Isokylä tunnel, Salo, Finland
For more information, please contact: Kirsi Hemmilä, Communications Manager, Caverion Finalnd, tel. 358 50 390 0941, kirsi.hemmila@caverion.com
Caverion and the Finnish Transport Agency have made an agreement on the modernisation of the telematic system for managing traffic on the E18 motorway between Turku and Muurla. The agreement also includes a major renovation of the Isokylä tunnel in Salo. Caverion is the main contractor of the project, and the value of the agreement is EUR 5.5 million.
"This modernisation project between Turku and Muurla is the first major replacement investment for telematics in Finland. Finnish Transport Agency is happy to have an experienced partner to carry out this challenging task," says Juha Sillanpää, Project Manager at Finnish Transport Agency.
Caverion will execute a project to modernise a telematic system, installed in 2003, over a 60-kilometre stretch of motorway. Work began in May and the target is to deploy the new system by the end of the year. At the same time, the concrete and steel structures and ventilation system in the 435-meter-long Isokylä tunnel will be repaired and the tunnel's safety systems will be modernised. The tunnel works will continue until midsummer.
"Digital functionality is a part of our modern-day mobility, and up-to-date control systems will have a major impact on transport and tunnel safety. A uniquely challenging aspect for the execution of this project is reconciling traffic diversions and lane closures with work on a busy road," says Ville Tamminen, Head of Division Finland at Caverion.
The Finnish Transport Agency belongs to Caverion's public sector client segment. Find out more about our services for public-sector infrastructure projects
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Wildfires Are Bad for Your Butter, Study Shows
Researchers found a significant decrease in milk production when dairy cows were exposed to smoke.
Last fall, when wildfires raged across parts of Oregon, more than two dozen dairy farms had to figure out how to milk and feed their cattle despite the dangerous conditions and poor air quality. One farmer in the Willamette Valley said that the heavy smoke meant that his cows had to be put in trailers, which both stressed them out and upended their daily routines.
Wildfires have unfortunately become a more regular occurrence in the western United States, and researchers are working to determine what other effects that smoke inhalation can have on dairy cattle, including what impact it can have on their milk production. Ashly Anderson, a graduate student at the University of Idaho College of Agricultural and Life Sciences, along with assistant professors Pedram Rezamand and Amy Skibiel took blood samples from 28 purebred Holstein cows and their calves last summer, to study whether certain immune markers changed before, during, or after the cows were exposed to a week of heavy smoke from the Snake River Complex fires.
Weed, California USA - June 29, 2021: The Lava Fire burns on the northwest side of Mount Shasta in Siskiyou County, California. Cows lined up look away from the wildfire. Trevor Bexon / Shutterstock
"Due to climate change and global conditions, we're going to be seeing a lot more wildfires and because of that there are going to be a lot more people and animals exposed to wildfires," Anderson said. "Being able to tell what kind of effects there are and how we might be affected in the future is very important."
According to High Country News, the University of Idaho researchers also analyzed five years of data from two dairy farms in Idaho and Washington state, along with the weather patterns and air quality from that same time span. In addition to the blood samples, measurements, and metabolic data, they also took note of the Holsteins' milk production for a three-month period last year, including during a "major weeklong smoke event."
Their data seem to show a "significant" decrease in milk production; the cows studied produced three pounds of milk — around one-third of a gallon — less every day. (High Country News reports that the average dairy cow in the United States produces 65 pounds of milk per day.) In addition to giving less milk, the cattle studied were more prone to udder infections and showed signs of inflammation. "This raises more questions," Skibiel told the outlet. "The connection between inhaling wildfire smoke and irritated udders is intriguing. It's certainly worth following up on."
Interestingly, an Oregon dairy farm whose Guernsey cows were not included in the study, said that she saw an increase in the number of calves that developed pneumonia in the weeks that followed the wildfires, and those cases "seemed to set in harder and faster than usual." And another Idaho rancher told the Idaho Capital Sun that the fires affected his pastures because wild game are coming onto his land to find food — and that, in turn, affects his cattle. Due to the decrease in "high-quality grass," he said his herd may have lower pregnancy rates this fall, and lower birth rates next spring.
"People are starting to pay attention and starting to realize wildfire issues are going to increase for the foreseeable future," Skibiel said. "I'm really hoping within the next year we have a better sense of the big picture."
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Today at the DevNation conference in San Francisco, Red Hat's Mark Little was joined on-stage by Alasdair Nottingham from IBM, Theresa Nguyen from Tomitribe, Mike Croft from Payara and Martijn Verburg from the London Java Community to announce a new community collaboration – MicroProfile – whose goal is to make it easier for developers to use familiar Java EE technologies and APIs for building microservice applications.
As organizations start to think about the next generation of those business-critical applications, many of them are likely thinking about cloud-native, Linux containers and microservices, and how they evolve by using the technologies and skills they already have.
Red Hat, IBM, Payara, Tomitribe and the London Java Community believe that Enterprise Java is a solid foundation on which to build the next generation of applications and the MicroProfile (which may ultimately become a submission for a standard specification) can make it easier and provide portability between vendor's implementations. The first release of the MicroProfile is expected to be available in September, with Red Hat's implementation to be based on WildFly Swarm.
Red Hat continues to support those in the Enterprise Java community that are working hard to move Enterprise Java forward by pushing ahead on the evolution of Java EE. To emphasize this point, Red Hat has underlined its support for Java EE 8 and is committed to finishing the JSRs it leads, like CDI 2.0, and any necessary enhancements to Bean Validation while it also invests in the MicroProfile. We see synergy between the current Enterprise Java community efforts and the newly announced MicroProfile, which is born out of the same Enterprise Java community. To us it's clear that the Enterprise Java community is forging ahead.
Red Hat understands that Enterprise Java has been successful for almost two decades thanks to the community collaboration that drove its evolution. Please join and participate in the MicroProfile effort and let's all take the next step forward by cooperatively innovating to bring the microservice architecture to Enterprise Java.


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Yesterday we asked you what you thought would be the top household items that consist of the most germs. If items found in the bathroom like a toothbrush holder or the bathroom door knob are at the top of your list, you're wrong!
But, don't despair, you weren't the only one who got it wrong.
NSF International, an independent public health and safety organisation, analysed 30 everyday household items to find out which contained the most germs. Many household members thought the bathroom items would be the dirtiest. However, based on the swab test by the NSF, of the 30 household items listed, the kitchen is where the dirtiest item is found.
The germiest item in the household is the very thing that is, ironically, used to clean dishes and kitchen counter tops— the dish sponge.
However if you answered toothbrush holder, you were close to being correct. The toothbrush holder comes in third place as being the germiest household item. The second was the kitchen sink.
Read more about this study here plus some tips on how to keep your household items and surfaces clean.