Air Drag and the Dependence on the Height of RidgesHakan Alpan, Jeffrey K. Landgren, Dr. Johna Leddy

Department of chemistry, University of Iowa

Introduction• Most models use a constant air drag coefficient.• Simplified one dimensional models, built around the conservation of

momentum equation.• Model 1 includes a constant coefficient of drag.• Model 2 includes a coefficient of drag that depends on height.• We assume left ice boundary attached to land and right boundary

has no stress because it is in the water.• Discretization is done to approximate our variables of interest.

The Model Results

• Significant difference in magnitude of stress.• Drag coefficient needs to include ridge height.• Improvements are needed.

Conclusion

• Conservation of momentum• Air drag• Wind drag• Coriolis effect• Internal stress• Pressure due to tilting

• Thermodynamics• Conservation of mass• Salt balance• Various feedback systems

• Melt pond feedback• Ice-albedo feedback

• Global warming increases the importance of modelling our climate.• Sea ice is a significant component to the climate.

Conservation of momentum

𝑚 ∗𝜕𝑣

𝜕𝑡= 𝜏𝑎 + 𝜏𝑤 −𝑚𝑓𝑐𝑘 × 𝑢 + 𝛻𝜎 −𝑚𝑔𝛻𝐻

General Equation

Parameters of Current Models

Forces acting on the ice

ReferencesAndreas, Edgar L, Manfred A. Lange, Stephen F. Ackley, and Peter Wadhams. "Roughness of Weddell Sea Ice and Estimates of the Air-ice Drag Coefficient." Journal of Geophysical Research 98.C7 (1993): 12439–12452. Wiley Online Library. Web. 20 July 2015. Hibler, W. D. "A Dynamic Thermodynamic Sea Ice Model." Journal of Physical Oceanography 9.4 (1979): 815-46. AMS Journals Online. AMS. Web. 20 July 2015. Laxon, Seymour, Neil Peacock, and Doug Smith. "High Interannual Variability of Sea Ice Thickness in the Arctic Region." Nature 425 (2003): 947-50. Nature. Nature. Web. 20 July 2015. Przybylak, Rajmund. The Climate of the Arctic. Dordrecht: Kluwer Academic, 2003. Print. Tsamados, Michel, Daniel L. Feltham, David Schroeder, Daniela Flocco, Sinead L. Farrell, Nathan Kurtz, Seymour W. Laxon, and Sheldon Bacon. "Impact of Variable Atmospheric and Oceanic Form Drag on Simulations of Arctic Sea Ice." Journal of Physical Oceanography 44.5 (2014): 1329-353. AMS Journals Online. AMS. Web. 20 July 2015.

Coriolis forceAcknowledgements

Secondary Student Training Program Dr. Johna Leddy Jeffrey Landgren Leddy Lab

Albedo

Simplified conservation of momentum

𝑚 ∗𝜕𝑣

𝜕𝑡= 𝜏𝑎 + 𝛻𝜎

Stress

𝜎 = 𝐸 ∗𝜕𝑢

𝜕𝑥Constant air drag𝜏𝑎 = 𝜌𝑎 ∗ 𝑐𝑎 ∗ 𝑣𝑎

2

Model 2 EquationsSimplified conservation of momentum

𝑚 ∗𝜕𝑣

𝜕𝑡= 𝜏𝑎 + 𝛻𝜎

Varying drag coefficient

𝑐𝑣𝑎 =1

2𝑐𝑟𝑎𝑆𝑐

2𝐻𝑠

𝐷𝑠𝐴(ln( 𝐻𝑠 𝑍0𝑖)

ln( 10 𝑍0𝑖))2

Sheltering function

𝑆𝑐 = (1 − 𝑒(−𝑠𝑙∗ 𝐷 𝐻))

Stress

𝜎 = 𝐸 ∗𝜕𝑢

𝜕𝑥

Varying air drag𝜏𝑎 = 𝜌𝑎 ∗ 𝑐𝑣𝑎 ∗ 𝑣𝑎

2

Key𝜌𝑎 and 𝜌𝑤 air and water drag densities 𝑚 mass of the ice𝑐𝑎 and 𝑐𝑤 air and water drag coefficients 𝑣 velocity function𝑓𝑐 Coriolis parameter 𝑐𝑟𝑎 local drag coefficient𝑍0𝑖 sea ice roughness parameter 𝑆𝑙 attenuation parameter𝐷 distance from the object 𝐻 height of the ice particle𝐻𝑠 and 𝐷𝑠 average of heights and distances of obstacles respectively

Model 1 Equations

Melt ponds

Motivation• Lack of precision from previous models.• Most models do not include air drag that depends on ice ridges.

Coriolis effect. Digital image. Coriolis Effect. N.p., n.d. Web. 20 July 2015N.d. The NPEO Web Cameras and Summer Melt Ponds. Web. 20 July 2015.

Ice-Albedo Feedback Loop. Digital image. Arctic Sea Ice at Minimum Extent Based on Satellite Record. N.p., 6 Oct. 2012. Web. 20 July 2015.

Sea Ice Extent. Digital image. Arctic Sea Ice Volume: PIOMAS, Prediction, and the Perils of Extrapolation. N.p., 11 Apr. 2012. Web. 20 July 2015.

Model 1 Model 2

Model 1 Model 2

• Initial time step shows difference in models.• Similar trends, but Model 2 is more precise.

t = 0.002 s

t = 50 s