.. _soil_energy_exchanges:

LST-driven soil energy balance
==============================

The LST-driven soil energy balance module namely involves (1) closure of the total soil energy input and outputs, (2) intra-soil energy exchanges mediated by thermal gradients and water fluxes, (3) an LST-forced surface-to-suburface energy heat flux submodule, and (4) diagnostic derivations of soil temperature and liquid H2O fraction, in coordination with the soil H2O balance module. 


.. image:: ../../images/CARDAMOM_TECHNOTE_ENERGY_BALANCE_SEP24.png


1. Closure of the soil energy inputs and outputs
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The soil energy balance is summarized as

.. math::

   \frac{dE}{dt} = E_{surface_in} +  E_{geology_in} - E_{H2O_in_out}

where \frac{dE}{dt} is the time-varying soil energy content, E_{surface_in} is the surface-to-subsurface gradient mediated thermal heat flux, E_{geology_in} is the upward thermal heat flux at the bottom boundary of the DALEC soil column, and E_{H2O_in_out} is the sum of all water-mediated input and output fluxes..

The three DALEC 1100 layers are specifically defined as 

.. math::

   \frac{dE1}{dt} = E_{surface_in} + E_{h2o_infil}  - E_{H2O_Q1} - E_{H2O_evap} - E_{H2O_transp1} - E_{LY1_2_LY2} - E_{H2O_LY1_2_LY2};

.. math::

   \frac{dE2}{dt} = E_{LY1_2_LY2} +E_{H2O_LY1_2_LY2} - E_{H2O_Q2}  - E_{H2O_transp2}  - E_{LY2_2_LY3} - E_{LY2_2_LY3} - E_{H2O_LY2_2_LY3};

.. math::

   \frac{dE2}{dt} = E_{geology_in} +  E_{LY2_2_LY3} + E_{H2O_LY2_2_LY3} - E_{h2o_Q3};






2. Water-mediated energy fluxes
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2. intra-soil energy exchanges mediated by thermal gradients and water fluxes
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The bulk soil energy inputs consist of (i) ground heat flux, (ii) inter-layer thermal exchanges, (iii) geological heat flux, and (iv) water-flux mediated energy gains and losses. The ground heat flux is modeled using the following formulation, where:

.. math::

   F_{ground\_heat} [PAUL REVIEW] = \kappa_{LY1} \frac{LY1 - LST}{0.5 d_{LY1}} 

Internal soil heat exchanges are simulated as

.. math::

   F_{exchange\{n,n+1\}} [PAUL REVIEW] = \sqrt{K_{LYn}K_{LYn+1}} \frac{T_{LYn+1} - T_{LYn}}{0.5(d_{LYn}+d_{LYn+1})} 

where :math:`K_{LYn}` represents the :math:`n^{th}` layer thermal conductivity.

Geological heat flux is assumed to be a constant input of 105 mW/m². Water-flux mediated energy exchanges are calculated based on the product of H₂O flux and temperature-based calculation of energy per unit volume H₂O. Infiltration-based inputs are assumed to be at 0°C temperature for snowmelt and max(T₂m, 0°C) for liquid precipitation inputs; subsurface runoff losses from layer :math:`n` are assumed to be at the corresponding layer temperature, :math:`T_{LYn}`.



3. LST-forced surface-to-suburface energy heat flux submodule
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4. diagnostic derivations of soil temperature and liquid H2O fraction, in coordination with the soil H2O balance module. 
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