<p>Glen's flow law is an empirical law derived from both field and experimental data (Glen 1955). In its original form Glen expressed the law as
<span class="math-container">$$
\dot\varepsilon = B \exp{(-Q/RT)}\sigma^n
$$</span>
where <span class="math-container">$\dot\varepsilon$</span> is the deformation rate, <span class="math-container">$B$</span> <span class="math-container">$Q$</span> and <span class="math-container">$n$</span> are empirical constants, <span class="math-container">$R$</span> is the gas constant and <span class="math-container">$T$</span> is the absolute temperature.
This formulation introduces a temperature dependence on the deformation.</p>
<p>The empirical constant <span class="math-container">$n$</span> is commonly taken as being <span class="math-container">$=3$</span> but this is basically an average of various experimental results. Goldsby (2009) showed that <span class="math-container">$n$</span> can be both smaller and larger depending if the stresses are very low or very high, respectively. Millstein et al. (2022) show that the stress dependence may be even larger than previously thought with <span class="math-container">$n\approx 4.1$</span> in fast flowing areas of ice sheets.</p>
<p>The viscosity parameter <span class="math-container">$A$</span> (<span class="math-container">$B$</span> in Glen's notation) will vary with ice temperature as seen above. However, the viscosity parameter is also sensitive to several other parameters such as pressure, crystal size and shape, crystal orientation, impurities in the ice (Gow and Williamson 1976, Hooke 1981, Alley 1992). Hence the viscosity parameter can be seen as site specific.</p>
<p>Clearly the value of B may be impossible to predict unless a wealth of information is known about the glacier ice throughout its body. So one way do deal with this is to use the viscosity parameter as a tuning parameter. Usually this involves comparing model output with measured velocities. This approach has been implemented in many different ways by modellers of glaciers and ice sheets. In models with a thermodynamic component the flow law may still include a temperature dependence.</p>
<p>In the review by Hooke (1981) it also seems clear that laboratory and field experiments yielding the empirical constants <span class="math-container">$A$</span> and <span class="math-container">$n$</span> diverge. One reason for this is that small laboratory specimens of ice yield values that are not fully representative of natural conditions.</p>
<p>So, the final comment is that using a single value of A is a last resort when the complexity of the natural ice cannot be accommodated in calculations or models. The value of <span class="math-container">$A$</span> is still site specific and not generally transferable between glaciers or sites on glaciers. In the end the use of solution for the flow law depends on what is &quot;good enough&quot; for the problem to be solved.</p>
<p>References</p>
<p>Alley R, 1992. Flow-law hypotheses for ice-sheet modeling. Journal of Glaciology, 38(129), 245-256. doi:10.3189/S0022143000003658</p>
<p>Glen JW, 1955. The creep of polycrystalline ice. Proceedings of the
Royal Society, Series A228 519–538. <a href="https://doi.org/10.1098/rspa.1955.0066" rel="nofollow noreferrer">https://doi.org/10.1098/rspa.1955.0066</a></p>
<p>Goldsby D, 2009. Superplastic flow of ice relevant to glacier and ice-sheet
mechanics. In Knight P, (ed.) Glacier Science and Environmental Change. Oxford: Wiley-Blackwell, 527pp. <a href="https://doi.org/10.1002/9780470750636.ch60" rel="nofollow noreferrer">https://doi.org/10.1002/9780470750636.ch60</a></p>
<p>Gow AJ and Williamson T, 1976. Rheological implications of the internal structure and crystal fabrics of the West Antarctic ice sheet as revealed by deep core drilling at Byrd Station. Geological Society of America Bulletin, 87,
1665–1677.</p>
<p>Hooke RLeB, 1981. Flow law for polycrystalline ice in glaciers: comparison of theoretical predictions, laboratory data, and field measurements. Reviews of Geophysics and Space Physics, 19(4), 664–672. <a href="https://doi.org/10.1029/RG019i004p00664" rel="nofollow noreferrer">https://doi.org/10.1029/RG019i004p00664</a></p>
<p>Millstein JD, Minchew BM and Pegler SS, 2022. Ice viscosity is more sensitive to stress than commonly assumed. Commun Earth Environ 3, 57. <a href="https://doi.org/10.1038/s43247-022-00385-x" rel="nofollow noreferrer">https://doi.org/10.1038/s43247-022-00385-x</a></p>

CC BY-SA 4.0
2023-05-05T14:39:00.620Z
2023-05-05T14:39:00.620Z
Peter Jansson
<p>I'm trying to better understand why I see so many people pick a fixed value of <span class="math-container">$A$</span> in Glen's Flow law when doing glacier modeling. I found that using a simple Arrhenius relationship we get the formula</p>
<p><span class="math-container">$A(T) = \left( A_0 exp\left(\frac{-Q}{R(T- \beta P)}\right) \right)$</span>   (from page 12 of <a href="https://www.researchgate.net/publication/267748557_Simulating_the_climatic_response_of_Hardangerjokulen_in_southern_Norway_since_the_Little_Ice_Age_Master%27s_thesis_in_Earth_Science" rel="nofollow noreferrer">this</a> source)</p>
<p>This formula would imply that temperature affects the value of <span class="math-container">$A$</span> so I would think that there would be a different <span class="math-container">$A$</span> at different depths within the glacier. However, I've also found out that we don't usually have temperature data at different depths within a glacier. This indicates to me that we can't use the above function to calculate <span class="math-container">$A(T)$</span> at different depths within the glacier.</p>
<p>This <a href="https://glaciers.gi.alaska.edu/sites/default/files/Notes_thermodynamics_Aschwanden.pdf" rel="nofollow noreferrer">source</a> (page 1) makes it seem like we just pick a type of thermal structure for the glacier at hand (cold, temperate, or polythermal) and then use that to determine what constant we should pick for <span class="math-container">$A$</span>.</p>
<ul>
<li>Why is fixing <span class="math-container">$A$</span> to be a constant seen as an accurate approach if <span class="math-container">$A$</span> depends so heavily on temperature? Does the thermal structure for the glacier at hand (cold, temperate, or polythermal) really give us enough information to accurately fix <span class="math-container">$A$</span> to be a constant value?</li>
</ul>
<p>Any authors or key phrases to look into for more information on this would be appreciated.</p>
<p>Any information is greatly appreciated. I'm very new to this.</p>

CC BY-SA 4.0
2023-05-03T19:45:22.163Z
The model of this site is to restrict each post to one question. You have several questions so it would be useful to divide the post into several posts to cover each question. Also, your "reference" is a lecture note. THere is need for better own research.
Peter Jansson
CC BY-SA 4.0
2023-05-04T00:09:31.303Z
@PeterJansson It was really all just one question about why A is taken to be constant and I listed out what confused me. But I've edited now so it appears there is just one solid question. As for the resource, personally, since it is a known formula I think lecture notes are fine. I'm new to this and teaching myself so if you have better suggestions for resources I would love to look at them.
k12345
<climate-models><glaciology>
Understanding Temperature's Effect on Glen's Flow Rate Factor A = A(T) and Why we Take A = Constant
k12345
# Understanding Temperature's Effect on Glen's Flow Rate Factor A = A(T) and Why we Take A = Constant

(Asked by: username_0 on 2023-05-02)

I'm trying to better understand why I see so many people pick a fixed value of $A$ in Glen's Flow law when doing glacier modeling. I found that using a simple Arrhenius relationship we get the formula

$A(T) = \left( A\_0 exp\left(\frac{-Q}{R(T- \beta P)}\right) \right)$ (from page 12 of [this](https://www.researchgate.net/publication/267748557_Simulating_the_climatic_response_of_Hardangerjokulen_in_southern_Norway_since_the_Little_Ice_Age_Master%27s_thesis_in_Earth_Science) source)

This formula would imply that temperature affects the value of $A$ so I would think that there would be a different $A$ at different depths within the glacier. However, I've also found out that we don't usually have temperature data at different depths within a glacier. This indicates to me that we can't use the above function to calculate $A(T)$ at different depths within the glacier.

This [source](https://glaciers.gi.alaska.edu/sites/default/files/Notes_thermodynamics_Aschwanden.pdf) (page 1) makes it seem like we just pick a type of thermal structure for the glacier at hand (cold, temperate, or polythermal) and then use that to determine what constant we should pick for $A$.

* Why is fixing $A$ to be a constant seen as an accurate approach if $A$ depends so heavily on temperature? Does the thermal structure for the glacier at hand (cold, temperate, or polythermal) really give us enough information to accurately fix $A$ to be a constant value?

Any authors or key phrases to look into for more information on this would be appreciated.

Any information is greatly appreciated. I'm very new to this.

**Tags:** climate-models, glaciology

> **Comments:**
> **username_1**: The model of this site is to restrict each post to one question. You have several questions so it would be useful to divide the post into several posts to cover each question. Also, your "reference" is a lecture note. THere is need for better own research.  
> **username_0**: @username_1 It was really all just one question about why A is taken to be constant and I listed out what confused me. But I've edited now so it appears there is just one solid question. As for the resource, personally, since it is a known formula I think lecture notes are fine. I'm new to this and teaching myself so if you have better suggestions for resources I would love to look at them.  

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## 1 answers

### Answer by username_1 on 2023-05-05. Score: 1

Glen's flow law is an empirical law derived from both field and experimental data (Glen 1955). In its original form Glen expressed the law as
$$
\dot\varepsilon = B \exp{(-Q/RT)}\sigma^n
$$
where $\dot\varepsilon$ is the deformation rate, $B$ $Q$ and $n$ are empirical constants, $R$ is the gas constant and $T$ is the absolute temperature.
This formulation introduces a temperature dependence on the deformation.

The empirical constant $n$ is commonly taken as being $=3$ but this is basically an average of various experimental results. Goldsby (2009) showed that $n$ can be both smaller and larger depending if the stresses are very low or very high, respectively. Millstein et al. (2022) show that the stress dependence may be even larger than previously thought with $n\approx 4.1$ in fast flowing areas of ice sheets.

The viscosity parameter $A$ ($B$ in Glen's notation) will vary with ice temperature as seen above. However, the viscosity parameter is also sensitive to several other parameters such as pressure, crystal size and shape, crystal orientation, impurities in the ice (Gow and Williamson 1976, Hooke 1981, Alley 1992). Hence the viscosity parameter can be seen as site specific.

Clearly the value of B may be impossible to predict unless a wealth of information is known about the glacier ice throughout its body. So one way do deal with this is to use the viscosity parameter as a tuning parameter. Usually this involves comparing model output with measured velocities. This approach has been implemented in many different ways by modellers of glaciers and ice sheets. In models with a thermodynamic component the flow law may still include a temperature dependence.

In the review by Hooke (1981) it also seems clear that laboratory and field experiments yielding the empirical constants $A$ and $n$ diverge. One reason for this is that small laboratory specimens of ice yield values that are not fully representative of natural conditions.

So, the final comment is that using a single value of A is a last resort when the complexity of the natural ice cannot be accommodated in calculations or models. The value of $A$ is still site specific and not generally transferable between glaciers or sites on glaciers. In the end the use of solution for the flow law depends on what is "good enough" for the problem to be solved.

References

Alley R, 1992. Flow-law hypotheses for ice-sheet modeling. Journal of Glaciology, 38(129), 245-256. doi:10.3189/S0022143000003658

Glen JW, 1955. The creep of polycrystalline ice. Proceedings of the
Royal Society, Series A228 519–538. <https://doi.org/10.1098/rspa.1955.0066>

Goldsby D, 2009. Superplastic flow of ice relevant to glacier and ice-sheet
mechanics. In Knight P, (ed.) Glacier Science and Environmental Change. Oxford: Wiley-Blackwell, 527pp. <https://doi.org/10.1002/9780470750636.ch60>

Gow AJ and Williamson T, 1976. Rheological implications of the internal structure and crystal fabrics of the West Antarctic ice sheet as revealed by deep core drilling at Byrd Station. Geological Society of America Bulletin, 87,
1665–1677.

Hooke RLeB, 1981. Flow law for polycrystalline ice in glaciers: comparison of theoretical predictions, laboratory data, and field measurements. Reviews of Geophysics and Space Physics, 19(4), 664–672. <https://doi.org/10.1029/RG019i004p00664>

Millstein JD, Minchew BM and Pegler SS, 2022. Ice viscosity is more sensitive to stress than commonly assumed. Commun Earth Environ 3, 57. <https://doi.org/10.1038/s43247-022-00385-x>


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