Sea ice 2

Lecture, exercises and practical by Jun.-Prof. Dr. Lars Kaleschke

• Monday 13:30-15:00
• Room ZMAW 022

Description of the course

The lecture will cover the thermodynamic coupling between the sea ice, the ocean, and the atmosphere. It is designed for master-level students with moderate knowledge in numerics, scientific programming, and sea ice physics. A conceptual model of the Arctic will be derived and simulation results will be analysed. For didactical reasons the model will be developed from scratch and kept as simple as possible, but complex enough to learn about the basic principles of the thermodynamic interaction between the ocean, the ice and the atmosphere for climatic, oceanographic and meteorological studies.

Acknowledgments

This lecture is based on content taken from a lecture Sea ice modeling by Aike Beckmann (Univ. Hamburg, Summer 2009) and a short course on Ice-Ocean Modeling and Data Assimilation which was conducted by Frank Kauker and Michael Karcher (Univ. Bremen, 6-7 December 2006).

Lesson 1 - Ocean mixed layer and radiative forcing without sea ice and atmosphere

Introduction and references for download

Scenario

• Ocean mixed layer forced by shortwave radiation only
• No atmosphere
• No exchange with deeper ocean layers, immediate mixing

• Heat balance at the sea surface: Short wave incoming radiation + long wave outgoing radiation

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• Numerics: forward-in-time integration, finite differences

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Research questions

Compute the time evolution of the ocean mixed layer temperature T_ml(t) for different h_ml, initial temperatures T_ml(t=0), and short wave insolation Q_SW.

Estimate the typical time scale for stationarity and select appropriate time step delta_t for model integration.

Change insolation after the model has reached stationarity.

Source code

/Gruppe1

Execute with ipython -pylab and run model1.py:

   1 from pylab import *
   2 #model1.py
   3 #LK 2010
   4 
   5 # model setup time
   6 iter=1000           # number of iterations
   7 dt=60*60*24.0       # time step one day [s]
   8 
   9 #model setup space 
  10 h_ml=10.            # oceanic mixed layer depth  [m]
  11 
  12 # physical constants
  13 T0=273.13           # zero degrees in [K]
  14 sig=5.67e-8         # Stefan-Boltzmann constant [W/m^2/K^4]
  15 # sea water
  16 albedo_sw=0.06     # short wave oceanic albedo
  17 eps_sw=0.97        # long wave oceanic emissivity 
  18 sig=5.67e-8         # Stefan-Boltzmann constant [W/m^2/K^4]
  19 rho_sw=1024.0       # sea water density 
  20 Cp_sw=4000.         # Oceanic heat capacity [J/Kg/K]
  21 
  22 #Forcing data
  23 I=400.0             # short wave insolation [W/m^2]
  24 
  25 #Prognostic variables
  26 T_ml=zeros(iter)    # oceanic mixed layer temperature [K]
  27 T_ml[0]=12.0  # initial temperature
  28 
  29 
  30 #Model integration
  31 for i in range(iter-1):
  32     Q1=(1-albedo_sw)*I # absorbed short wave radiation
  33     Q2=-eps_sw*sig*(T_ml[i]+T0)**4 # emitted long wave radiation
  34     
  35     T_ml[i+1]=T_ml[i]+(((Q1+Q2)*dt)/(h_ml*Cp_sw*rho_sw))
  36 
  37     if i==500: # Change forcing data after 600 days
  38         I=380.0
  39 
  40 # Display results
  41 figure(1)
  42 plot(T_ml)
  43 xlabel('Time [days]')
  44 ylabel(r'Temperature $T_{ml}$ [$^\circ$ C]')
  45 text(50,12.5,r'Initial temperature $T_{ml}=12^\circ$C')
  46 text(50,12.7,r'Short wave insolation $I=400$ W/m$^2$')
  47 text(500,14.0,r'Short wave insolation $I=380$ W/m$^2$')
  48 savefig('lesson1_results1.png',dpi=75)
  49 
  50 # Display 1/e time
  51 T_ml_scaled=(T_ml[501:999]-T_ml[999])/(T_ml[501]-T_ml[999]) # Normalize to 0-1
  52 figure(2)
  53 plot(T_ml_scaled)
  54 grid()
  55 axis('tight')
  56 yticks((1,1/e),('1','1/e'))
  57 xlabel('Time after day 500 [days]')
  58 ylabel(r'Normalized temperature $T_{ml}^*$')
  59 savefig('lesson1_results2.png',dpi=75)
  60 show()

Simulation results

Literature

Maykut, G.A. & N. Untersteiner, 1971: Some results from a time-dependent thermodynamic model of sea ice. J. Geophys. Res.,76, 1550-1575.

Semtner, A., 1976: A model for the thermodynamic growth of sea ice in numerical investigations of climate, J. Phys. Oceanogr, 6, 379-389.

Hibler III, W.D., 1979: A dynamic-thermodynamic sea ice model. J. Phys. Oceanogr., 9, 815-846.

Parkinson, C.L. & W.M. Washington, 1979: A large-scale numerical model of sea ice., J. Geophys. Res., 84, 311-337.