tREADME.rst - pism - [fork] customized build of PISM, the parallel ice sheet model (tillflux branch)
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tREADME.rst (7979B)
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1 .. default-role:: math
2 .. |prs| replace:: *Phillips et al*
3 .. |chs| replace:: cryo-hydrologic system
4
5 One-column cryo-hydrologic warming setup
6 ========================================
7
8 This directory contains a set up of the one-column cryo-hydrologic warming experiment
9 inspired by **Phillips, T. and Rajaram, H. and Steffen, K.**, *Cryo-hydrologic warming: A
10 potential mechanism for rapid thermal response of ice sheets*,
11 (https://doi.org/10.1029/2010GL044397).
12
13 Summary
14 -------
15
16 In summary: in ablation areas the water generated during the melt season drains to the
17 base through the channels in the |chs|. The presence of water
18 in the |chs| leads to warming of the ice adjacent to the channels.
19
20 The authors assume that the additional heat flux into the ice due to this mechanism is
21 proportional to the difference in temperature between the |chs| and the ice next to it:
22
23 .. math::
24
25 Q = \frac{k_{i}}{R^2} (T_{CH} - T_i)
26
27 Here `k_i` is the thermal conductivity of ice and `R` is the spacing between the
28 channels in the |chs| (a poorly-constrained model parameter).
29
30 |prs| model `T_i` using a familiar temperature-based energy balance equation
31 with a parameterization of cooling due to horizontal advection. The flux `Q` defined above
32 appears as an additional source term:
33
34 .. math::
35
36 \frac{\partial \rho_i C_i T_i}{\partial t} + \frac{\partial (\rho_i C_i w T_i)}{\partial z}
37 - \frac{\partial}{\partial z}\left(k_i \frac{\partial T_i}{\partial z}\right)
38 = - \frac{\partial \rho_i C_i u T_i}{\partial x} + Q_{s} + Q
39
40 Here `\rho_i` is the density of ice, `u` and `w` are horizontal and vertical velocity
41 components, `C_i` is the specific heat capacity of ice, and `Q_s` is the strain heating.
42
43 The temperature in the |chs| is modeled using an enthalpy-based energy equation omitting
44 advection terms:
45
46 .. math::
47
48 \frac{\partial \bar{\rho H}}{\partial t}
49 - \frac{\partial}{\partial z}\left(\bar{k} \frac{\partial T_{CH}}{\partial z}\right)
50 = - Q
51
52 Here the enthalpy is defined by
53
54 .. math::
55 :name: enthalpy
56
57 \rho H = (1 - \phi_w) \rho_i C_i T_{CH} + \phi_w (\rho_w C_w T_{CH} + L),
58
59 where `\phi_w` is the water fraction, `L` is the latent heat of fusion, `\rho_w`, is the
60 water density, and `C_w` is the specific heat capacity of fresh water.
61
62 The thermal conductivity of the mixture, `\bar k`, is defined by
63
64 .. math::
65
66 \bar k = (1 - \phi_w) k_i + \phi_w k_w.
67
68 The authors use Dirichlet (temperature) boundary conditions at the top boundary (assuming
69 that the temperature at the top of the snow or ice is equal to the near-surface air
70 temperature, but never exceeds the pressure melting point) and Dirichlet or Neumann
71 (mentioned but not specified in the paper) at the bottom boundary.
72
73 It is also mentioned that they account for *snow cover, which insulates the ice surface
74 from cold winter temperatures* and *temperature dependence of the thermal properties was
75 incorporated in the model*, but no details are provided.
76
77 During the melt season (when the water is present) the temperature in the |chs| is set to
78 the pressure-melting and the water fraction is set to a predetermined constant (|prs| use
79 `0.005`, i.e. one half of a percent) while during the winter the |chs| is allowed to cool.
80
81 Comparing to PISM's energy conservation model
82 ---------------------------------------------
83
84 - PISM's definition of enthalpy is equivalent to the one above, with some caveats. (The
85 equation in |prs| has a couple of issues that I don't need to go into here.)
86
87 Note that in PISM's definition the temperature never exceeds `T_m(p)`, the
88 pressure-melting point, while in |prs| `T_{CH} > T_m(p)` is allowed, but `T_{CH}` never
89 exceeds `T_m(p)` due to the setup they use.
90
91 - PISM's energy conservation model includes all the mechanisms modeled by equations above
92 *except* for the parameterization of advection-driven cooling not necessary in a 3D model.
93
94 - PISM's enthalpy-based energy conservation model can be used to model *both* the ice and
95 the |chs| columns.
96
97 - PISM assumes that the specific heat capacity of ice is constant. The thermal
98 conductivity can be a function of temperature, but does not depend on the water fraction
99 (unlike `\bar k` above). Note, though, that for water fractions at or below `0.005` the
100 value of `\bar k` is very close to `k_i`.
101
102 With default parameter values PISM "ignores" thermal conductivity of temperate ice
103 *without* completely removing it: this can be thought of as a *regularization*.
104 Specifically, the thermal conductivity of temperate ice is set to a given fraction of
105 `k_i` (see ``energy.enthalpy.temperate_ice_thermal_conductivity_ratio``).
106
107 This one-column setup uses ``energy.ch_warming.temperate_ice_thermal_conductivity_ratio``
108 (equal to `1`) to get us close to the equation modeling |chs| enthalpy evolution in
109 *Phillips et al*.
110
111 Technical details
112 -----------------
113
114 This setup models a 200-meter deep ice column using an equally-spaced vertical grid with a
115 1-meter resolution (201 grid points) and takes equal time steps 1 day long each.
116
117 The flux `Q` from the |chs| into the ice is added to the deformational strain heating term
118 in PISM's energy balance equation.
119
120 Ice velocity components are set to zero, eliminating advection terms and resulting in zero
121 strain heating due to ice flow.
122
123 We use a Dirichlet boundary condition at the top surface
124
125 .. math::
126
127 \begin{aligned}
128 H_{\text{surface}} &=
129 \begin{cases}
130 H(T_{\text{surface}}), & T_{\text{surface}} < T_m, \\
131 H(T_m), & T_{\text{surface}} \ge T_m,
132 \end{cases}\\
133 T_{\text{surface}} &= T_{\text{mean}} + A\cdot \cos(2 \pi t - \phi_0)
134 \end{aligned}
135
136 with time `t` in years. The values of `A` (`6 K`) and `T_{\text{mean}}` (`-5^{\circ} C`)
137 are selected to get a 9-week-long melt season.
138
139 We use a Neumann boundary condition at the bottom surface:
140
141 .. math::
142
143 \left.\frac{\partial H}{\partial z}\right|_{z=0} = - \frac{G}{k_i},
144
145 where `G` is the upward geothermal heat flux, `W / m^2`. This setup uses `G = 0`.
146
147 Given the absence of ice flow, we use the steady-state enthalpy distribution corresponding
148 to the mean-annual temperature at the surface and the constant heat flux at the base of
149 the ice:
150
151 .. math::
152
153 \begin{aligned}
154 H_{\text{steady state}} &= H(T_{\text{steady state}}, p)\\
155 T_{\text{steady state}} &= (h - z) \frac{G}{k_i}\\
156 \end{aligned}
157
158 where `h` is ice thickness and `p` is the hydrostatic pressure. We assume that the mean
159 annual air temperature and ice thickness are low enough so that this does not result in a
160 layer of temperate ice near the base.
161
162 At the beginning of each time step the enthalpy in the |chs| is set to `H(T_m(p),
163 \omega_0)` if `T_{\text{surface}} \ge T_m` (here `\omega_0 = 0.005`), simulating the
164 presence of liquid water in the |chs| during the melt season.
165
166 .. note::
167
168 One could use the surface mass balance to detect the melt season instead:
169 `T_{\text{surface}}` may not capture the daily temperature variability.
170
171 Results
172 -------
173
174 This annual air temperature cycle results in a 9-week-long melt season.
175
176 .. figure:: air-temperature.png
177
178 Air temperature
179
180 .. figure:: ice-temperature.png
181
182 Ice temperature in the top 15 meters of ice. Note the gradual warming with almost no
183 inter-annual variability at the depth of 15 meters.
184
185 .. figure:: ice-temperature-curves.png
186
187 Ice temperature evolution at different depths.
188
189 .. figure:: ch-temperature.png
190
191 Temperature in the |chs|. The end of the melt season can be seen
192 clearly.
193
194 .. figure:: ch-water-fraction.png
195
196 Water fraction in the |chs|. Note that at higher depths the water fraction stays above
197 zero all the way through the winter, so the temperature in the |chs| never drops below
198 pressure-melting.
199
200 Next steps
201 ----------
202
203 - Adjust the value of `R` and observe the changes in the evolution of ice temperature.
204 - Compare to **Hills et al**, *Processes influencing near-surface heat transfer in
205 Greenland’s ablation zone*, (https://doi.org/10.5194/tc-2018-51).