From 673d808bbc47b1a468a01fc496fa2aa550b0a5b5 Mon Sep 17 00:00:00 2001 From: Naoki Mizukami Date: Wed, 6 May 2026 14:26:27 -0600 Subject: [PATCH 01/11] A few minor edits and various corrections e.g., link, equations. etc. --- .../tech_note/Lake/CLM50_Tech_Note_Lake.rst | 65 +++++++++---------- 1 file changed, 29 insertions(+), 36 deletions(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 88cb77d737..cae9fef802 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -3,14 +3,14 @@ Lake Model ============= -The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `, :ref:`Hostetler et al. (1993) `. Lakes have spatially variable depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`Surface Albedo Lake`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. +The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `. Lakes have spatially varying depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`Surface Albedo Lake`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. .. _Vertical Discretization Lake: Vertical Discretization --------------------------- -Currently, there is one lake modeled in each grid cell (with prescribed or assumed depth *d*, extinction coefficient :math:`\eta`, and fetch *f*), although this could be modified with changes to the CLM subgrid decomposition algorithm in future model versions. As currently implemented, the lake consists of 0-5 snow layers; water and ice layers (10 for global simulations and 25 for site simulations) comprising the "lake body;" 10 "soil" layers; and 5 bedrock layers. Each lake body layer has a fixed water mass (set by the nominal layer thickness and the liquid density), with frozen mass-fraction *I* a state variable. Resolved snow layers are present if the snow thickness :math:`z_{sno} \ge s_{\min }`, where *s*\ :sub:`min` = 4 cm by default, and is adjusted for model timesteps other than 1800 s in order to maintain numerical stability (section :numref:`Modifications to Snow Layer Logic Lake`). For global simulations with 10 body layers, the default (50 m lake) body layer thicknesses are given by: :math:`\Delta z_{i}` of 0.1, 1, 2, 3, 4, 5, 7, 7, 10.45, and 10.45 m, with node depths :math:`z_{i}` located at the center of each layer (i.e., 0.05, 0.6, 2.1, 4.6, 8.1, 12.6, 18.6, 25.6, 34.325, 44.775 m). For site simulations with 25 layers, the default thicknesses are (m): 0.1 for layer 1; 0.25 for layers 2-5; 0.5 for layers 6-9; 0.75 for layers 10-13; 2 for layers 14-15; 2.5 for layers 16-17; 3.5 for layers 18-21; and 5.225 for layers 22-25. For lakes with depth *d* :math:`\neq` 50 m and *d* :math:`\ge` 1 m, the top layer is kept at 10 cm and the other 9 layer thicknesses are adjusted to maintain fixed proportions. For lakes with *d* :math:`<` 1 m, all layers have equal thickness. Thicknesses of snow, soil, and bedrock layers follow the scheme used over non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications to the snow layer thickness rules to keep snow layers at least as thick as *s*\ :sub:`min` (section :numref:`Modifications to Snow Layer Logic Lake`). +Currently, there is one lake modeled in each grid cell (with prescribed or default depth *d*, extinction coefficient :math:`\eta`, and fetch *f*), although this could be modified with changes to the CLM subgrid decomposition algorithm in future model versions. As currently implemented, the lake consists of 0-5 snow layers; water and ice layers (10 for global simulations and 25 for site simulations) comprising the "lake body;" 10 "soil" layers; and 5 bedrock layers. Each lake body layer has a fixed water mass (set by the nominal layer thickness and the liquid density), with frozen mass-fraction *I* a state variable. Resolved snow layers are present if the snow thickness :math:`z_{sno} \ge s_{\min }`, where *s*\ :sub:`min` = 4 cm by default, and is adjusted for model timesteps other than 1800 s in order to maintain numerical stability (section :numref:`Modifications to Snow Layer Logic Lake`). For global simulations with 10 body layers, the default (50 m lake) body layer thicknesses are given by: :math:`\Delta z_{i}` of 0.1, 1, 2, 3, 4, 5, 7, 7, 10.45, and 10.45 m, with node depths :math:`z_{i}` located at the center of each layer (i.e., 0.05, 0.6, 2.1, 4.6, 8.1, 12.6, 18.6, 25.6, 34.325, 44.775 m). For site simulations with 25 layers, the default thicknesses are (m): 0.1 for layer 1; 0.25 for layers 2-5; 0.5 for layers 6-9; 0.75 for layers 10-13; 2 for layers 14-15; 2.5 for layers 16-17; 3.5 for layers 18-21; and 5.225 for layers 22-25. For lakes with depth *d* :math:`\neq` 50 m and *d* :math:`\ge` 1 m, the top layer is kept at 10 cm and the other 9 layer thicknesses are adjusted to maintain fixed proportions. For lakes with *d* :math:`<` 1 m, all layers have equal thickness. Thicknesses of snow, soil, and bedrock layers follow the scheme used over non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications to the snow layer thickness rules to keep snow layers at least as thick as *s*\ :sub:`min` (section :numref:`Modifications to Snow Layer Logic Lake`). .. _External Data Lake: @@ -56,7 +56,7 @@ Surface Properties The fraction of shortwave radiation absorbed at the surface, :math:`\beta`, depends on the lake state. If resolved snow layers are present, then :math:`\beta` is set equal to the absorption fraction predicted by the snow-optics submodel (Chapter :numref:`rst_Surface Albedos`) for the top snow layer. Otherwise, :math:`\beta` is set equal to the near infrared fraction of the shortwave radiation reaching the surface simulated by the atmospheric model or atmospheric data model used for offline simulations (Chapter :numref:`rst_Land-only Mode`). The remainder of the shortwave radiation fraction (1 :math:`{-}` :math:`\beta`) is absorbed in the lake body or soil as described in section :numref:`Radiation Penetration`. -The surface roughnesses are functions of the lake state and atmospheric forcing. +The surface roughnesses are functions of the lake state and atmospheric forcing. For unfrozen lakes (:math:`T_{g} > T_{f}`), :math:`z_{0m}` is given by (:ref:`Subin et al. (2012a) `) @@ -68,7 +68,7 @@ For unfrozen lakes (:math:`T_{g} > T_{f}`), :math:`z_{0m}` is given by (:ref:`Su where :math:`\alpha` = 0.1, :math:`\nu` is the kinematic viscosity of air given below, *C* is the effective Charnock coefficient given below, :math:`u_{*}` is the friction velocity (m/s), and *g* is the acceleration of gravity (:numref:`Table Physical Constants`). The kinematic viscosity is given by .. math:: - :label: 12.4 + :label: 12.4 \nu =\nu _{0} \left(\frac{T_{g} }{T_{0} } \right)^{1.5} \frac{P_{0} }{P_{ref} } @@ -79,7 +79,7 @@ where :math:`P_{ref}` is the pressure at the atmospheric reference height. The Charnock coefficient *C* is a function of the lake fetch *F* (m), given in the surface data or set to 25 times the lake depth *d* by default: .. math:: - :label: 12.5 + :label: 12.5 \begin{array}{l} {C=C_{\min } +(C_{\max } -C_{\min } )\exp \left\{-\min \left(A,B\right)\right\}} \\ {A={\left(\frac{Fg}{u_{*} ^{2} } \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right.} 3} } \mathord{\left/ {\vphantom {\left(\frac{Fg}{u_{*} ^{2} } \right)^{{1\mathord{\left/ {\vphantom {1 3}} \right.} 3} } f_{c} }} \right.} f_{c} } } \\ {B=\varepsilon \frac{\sqrt{dg} }{u} } \end{array} @@ -104,7 +104,7 @@ For frozen lakes ( :math:`T_{g} \le T_{f}` ) without resolved snow layers ( :math:`snl = 0` ), :math:`z_{0m} =z_{0m_{ice}} =2.3\times 10^{-3} {\rm m}` (:ref:`Meier et al. (2022) `). For frozen lakes with resolved -snow layers ( :math:`snl > 0` ), the momentum roughness length is evaluated based on accumulated snow melt :math:`M_{a} {\rm }` (:ref:`Meier et al. (2022) `). +snow layers ( :math:`snl > 0` ), the momentum roughness length is evaluated based on accumulated snow melt :math:`M_{a} {\rm }` (:ref:`Meier et al. (2022) `). For :math:`M_{a} >=1\times 10^{-5}` .. math:: @@ -123,7 +123,7 @@ For :math:`M_{a} <1\times 10^{-5}` Accumulated snow melt :math:`M_{a}` at the current time step :math:`t` is defined as .. math:: - :label: 12.5d + :label: 12.5d M ^{t}_{a} = M ^{t-1}_{a} - (q ^{t}_{sno} \Delta t + q ^{t}_{snowmelt} \Delta t)\times 10^{-3} @@ -169,7 +169,7 @@ where :math:`\vec{S}_{g}` \ is the absorbed solar radiation in the lake, .. math:: :label: 12.8 - \lambda =\left\{\begin{array}{l} {\lambda _{sub} \qquad T_{g} \le T_{f} } \\ {\lambda _{vap} \qquad T_{g} >T_{f} } \end{array}\right\}. + \lambda =\Bigg\{\begin{array}{l} {\lambda _{sub} \qquad T_{g} \le T_{f} } \\ {\lambda _{vap} \qquad T_{g} >T_{f} } \end{array} The sensible heat flux (W m\ :sup:`-2`) is @@ -241,7 +241,7 @@ where :math:`L_{g} \, \uparrow` is the upward longwave radiation from the surfac .. math:: :label: 12.16 - L\, \uparrow =\left(1-\varepsilon _{g} \right)L_{atm} \, \downarrow +\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{4} +4\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{3} \left(T_{g}^{n+1} -T_{g}^{n} \right) + \vec{L}_{g}\, \uparrow =\left(1-\varepsilon _{g} \right)L_{atm} \, \downarrow +\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{4} +4\varepsilon _{g} \sigma \left(T_{g}^{n} \right)^{3} \left(T_{g}^{n+1} -T_{g}^{n} \right) where :math:`\varepsilon _{g} =0.97` is the lake surface emissivity, :math:`\sigma` is the Stefan-Boltzmann constant (W m\ :sup:`-2` K\ :sup:`-4`) (:numref:`Table Physical Constants`), and @@ -285,29 +285,29 @@ where the partial derivatives are The fluxes of momentum, sensible heat, and water vapor are solved for simultaneously with lake surface temperature as follows. To begin, :math:`z_{0m}` and the scalar roughness lengths are set as described in section :numref:`Surface Properties Lake`. -#. An initial guess for the wind speed :math:`V_{a}` including the convective velocity :math:`U_{c}` is obtained from :eq:`5.24` assuming an initial convective velocity :math:`U_{c} =0` m s\ :sup:`-1` for stable conditions (:math:`\theta _{v,\, atm} -\theta _{v,\, s} \ge 0` as evaluated from :eq:`5.50`) and :math:`U_{c} =0.5` for unstable conditions (:math:`\theta _{v,\, atm} -\theta _{v,\, s} <0`). +#. An initial guess for the wind speed :math:`V_{a}` including the convective velocity :math:`U_{c}` is obtained from equation :eq:`5.24` assuming an initial convective velocity :math:`U_{c} =0` m s\ :sup:`-1` for stable conditions (:math:`\theta _{v,\, atm} -\theta _{v,\, s} \ge 0` as evaluated from equation :eq:`5.50`) and :math:`U_{c} =0.5` for unstable conditions (:math:`\theta _{v,\, atm} -\theta _{v,\, s} <0`). -#. An initial guess for the Monin-Obukhov length :math:`L` is obtained from the bulk Richardson number using :eq:`5.46` and :eq:`5.48`. +#. An initial guess for the Monin-Obukhov length :math:`L` is obtained from the bulk Richardson number using equations :eq:`5.46` and :eq:`5.48`. #. The following system of equations is iterated four times: -#. Heat of vaporization / sublimation :math:`\lambda` \ (:eq:`12.8`) +#. Heat of vaporization / sublimation :math:`\lambda` \ (equation :eq:`12.8`) #. Thermal conductivity :math:`\lambda _{T}` \ (above) -#. Friction velocity :math:`u_{*}` (:eq:`5.32`, :eq:`5.33`, :eq:`5.34`, :eq:`5.35`) +#. Friction velocity :math:`u_{*}` (equations :eq:`5.32`, :eq:`5.33`, :eq:`5.34`, :eq:`5.35`) -#. Potential temperature scale :math:`\theta _{*}` (:eq:`5.37`, :eq:`5.38`, :eq:`5.39`, :eq:`5.40`) +#. Potential temperature scale :math:`\theta _{*}` (equations :eq:`5.37`, :eq:`5.38`, :eq:`5.39`, :eq:`5.40`) -#. Humidity scale :math:`q_{*}` (:eq:`5.41`, :eq:`5.42`, :eq:`5.43`, :eq:`5.44`) +#. Humidity scale :math:`q_{*}` (equations :eq:`5.41`, :eq:`5.42`, :eq:`5.43`, :eq:`5.44`) -#. Aerodynamic resistances :math:`r_{am}`, :math:`r_{ah}`, and :math:`r_{aw}` (:eq:`5.55`, :eq:`5.56`, :eq:`5.57`) +#. Aerodynamic resistances :math:`r_{am}`, :math:`r_{ah}`, and :math:`r_{aw}` (equations :eq:`5.55`, :eq:`5.56`, :eq:`5.57`) -#. Lake surface temperature :math:`T_{g}^{n+1}` (:eq:`12.18`) +#. Lake surface temperature :math:`T_{g}^{n+1}` (equation :eq:`12.18`) -#. Heat of vaporization / sublimation :math:`\lambda` (:eq:`12.8`) +#. Heat of vaporization / sublimation :math:`\lambda` (equation :eq:`12.8`) -#. Sensible heat flux :math:`H_{g}` is updated for :math:`T_{g}^{n+1}` (:eq:`12.9`) +#. Sensible heat flux :math:`H_{g}` is updated for :math:`T_{g}^{n+1}` (equation :eq:`12.9`) #. Water vapor flux :math:`E_{g}` is updated for :math:`T_{g}^{n+1}` as @@ -320,11 +320,11 @@ where the last term on the right side of equation :eq:`12.23` is the change in s #. Saturated specific humidity :math:`q_{sat}^{T_{g} }` and its derivative :math:`\frac{dq_{sat}^{T_{g} } }{dT_{g} }` are updated for :math:`T_{g}^{n+1}` (section :numref:`Monin-Obukhov Similarity Theory`). -#. Virtual potential temperature scale :math:`\theta _{v*}` (:eq:`5.17`) +#. Virtual potential temperature scale :math:`\theta _{v*}` (equation :eq:`5.17`) -#. Wind speed including the convective velocity, :math:`V_{a}` (:eq:`5.24`) +#. Wind speed including the convective velocity, :math:`V_{a}` (equation :eq:`5.24`) -#. Monin-Obukhov length :math:`L` (:eq:`5.49`) +#. Monin-Obukhov length :math:`L` (equation :eq:`5.49`) #. Roughness lengths (section :numref:`Surface Properties Lake`). @@ -375,28 +375,21 @@ where :math:`\tilde{c}_{v}` is the volumetric heat capacity (J m\ :sup:`-3` K\ : \frac{d}{dt} \sum _{j=1}^{N}\left[\tilde{c}_{v,j} (t)\left(T_{j} -T_{f} \right)+L_{j} (t)\right] \Delta z_{j} =G+\left(1-\beta \right)\vec{S}_{g} -where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Radiation Penetration`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing eq. to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in eq.. - -.. _Overview of Changes from CLM4 2: - -Overview of Changes from CLM4 -^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ - -Thermal conductivities include additional eddy diffusivity, beyond the :ref:`Hostetler and Bartlein (1990)` formulation, due to unresolved processes (:ref:`Fang and Stefan 1996`; :ref:`Subin et al. (2012a) `). Lake water is now allowed to freeze by an arbitrary fraction for each layer, which releases latent heat and changes thermal properties. Convective mixing occurs for all lakes, even if frozen. Soil and bedrock are included beneath the lake. The full snow model is used if the snow thickness exceeds a threshold; if there are resolved snow layers, radiation transfer is predicted by the snow-optics submodel (Chapter :numref:`rst_Surface Albedos`), and the remaining radiation penetrating the bottom snow layer is absorbed in the top layer of lake ice; conversely, if there are no snow layers, the solar radiation penetrating the bottom lake layer is absorbed in the top soil layer. The lakes have variable depth, and all physics is assumed valid for arbitrary depth, except for a depth-dependent enhanced mixing (section :numref:`Eddy Diffusivity and Thermal Conductivities`). Finally, a previous sign error in the calculation of eddy diffusivity (specifically, the Brunt-Väisälä frequency term; eq. ) was corrected. +where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Heat Capacities Lake`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing eq. to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in eq.. .. _Boundary Conditions Lake: Boundary Conditions ^^^^^^^^^^^^^^^^^^^^^^^^^ -The top boundary condition, imposed at the top modeled layer :math:`i=j_{top}`, where :math:`j_{top} =-n_{sno} +1`, is the downwards surface flux *G* defined by the energy flux residual during the surface temperature solution (section :numref:`Boundary Conditions Lake`). The bottom boundary condition, imposed at :math:`i=N_{levlak} +N_{levgrnd}`, is zero flux. The 2-m windspeed :math:`u_{2}` \ (m s\ :sup:`-1`) is used in the calculation of eddy diffusivity: +The top boundary condition, imposed at the top modeled layer :math:`i=j_{top}`, where :math:`j_{top} =-n_{sno} +1`, is the downwards surface flux *G* defined by the energy flux residual during the surface temperature solution (section :numref:`Surface Flux Solution Lake`). The bottom boundary condition, imposed at :math:`i=N_{levlak} +N_{levgrnd}`, is zero flux. The 2-m windspeed :math:`u_{2}` \ (m s\ :sup:`-1`) is used in the calculation of eddy diffusivity: .. math:: :label: 12.28 u_{2} =\frac{u_{*} }{k} \ln \left(\frac{2}{z_{0m} } \right)\ge 0.1. -where :math:`u_{*}` \ is the friction velocity calculated in section :numref:`Boundary Conditions Lake` and +where :math:`u_{*}` \ is the friction velocity calculated in section :numref:`Monin-Obukhov Similarity Theory` and *k* is the von Karman constant (:numref:`Table Physical Constants`). .. _Eddy Diffusivity and Thermal Conductivities: @@ -419,7 +412,7 @@ where :math:`\kappa _{e}` is due to wind-driven eddies (:ref:`Hostetler and Bart .. math:: :label: 12.30 - m_{d} =\left\{\begin{array}{l} {1,\qquad d<25{\rm m}} \\ {10,\qquad d\ge 25{\rm m}} \end{array}\right\} + m_{d} =\Bigg\{\begin{array}{rr} {1,\qquad d<25{\rm m}} \\ {10,\qquad d\ge 25{\rm m}} \end{array} where *d* is the lake depth. @@ -428,7 +421,7 @@ The wind-driven eddy diffusion coefficient :math:`\kappa _{e,\, i}` (m\ :sup:`2` .. math:: :label: 12.31 - \kappa _{e,\, i} =\left\{\begin{array}{l} {\frac{kw^{*} z_{i} }{P_{0} \left(1+37Ri^{2} \right)} \exp \left(-k^{*} z_{i} \right)\qquad T_{g} >T_{f} } \\ {0\qquad T_{g} \le T_{f} } \end{array}\right\} + \kappa _{e,\, i} =\Bigg\{\begin{array}{rr} {\frac{kw^{*} z_{i} }{P_{0} \left(1+37Ri^{2} \right)} \exp \left(-k^{*} z_{i} \right)\qquad T_{g} >T_{f} } \\ {0\qquad T_{g} \le T_{f} } \end{array} where :math:`P_{0} =1` is the neutral value of the turbulent Prandtl number, :math:`z_{i}` is the node depth (m), the surface friction velocity (m s\ :sup:`-1`) is :math:`w^{*} =0.0012u_{2}`, and :math:`k^{*}` varies with latitude :math:`\phi` as :math:`k^{*} =6.6u_{2}^{-1.84} \sqrt{\left|\sin \phi \right|}`. For the bottom layer, :math:`\kappa _{e,\, N_{levlak} } =\kappa _{e,N_{levlak} -1\, }`. As in :ref:`Hostetler and Bartlein (1990)`, the 2-m wind speed :math:`u_{2}` (m s\ :sup:`-1`) (eq. ) is used to evaluate :math:`w^{*}` and :math:`k^{*}` rather than the 10-m wind used by :ref:`Henderson-Sellers (1985) `. @@ -492,7 +485,7 @@ The thermal conductivity of snow, soil, and bedrock layers above and below the l Radiation Penetration ^^^^^^^^^^^^^^^^^^^^^^^^^^^ -If there are no resolved snow layers, the surface absorption fraction :math:`\beta` is set according to the near-infrared fraction simulated by the atmospheric model. This is apportioned to the surface energy budget (section :numref:`Surface Properties Lake`), and thus no additional radiation is absorbed in the top :math:`z_{a}` (currently 0.6 m) of unfrozen lakes, for which the light extinction coefficient :math:`\eta` (m\ :sup:`-1`) varies between lake columns (eq. ). For frozen lakes (:math:`T_{g} \le T_{f}` ), the remaining :math:`\left(1-\beta \right)\vec{S}_{g}` fraction of surface absorbed radiation that is not apportioned to the surface energy budget is absorbed in the top lake body layer. This is a simplification, as lake ice is partially transparent. If there are resolved snow layers, then the snow optics submodel (Chapter :numref:`rst_Surface Albedos`) is used to calculate the snow layer absorption (except for the absorption predicted for the top layer by the snow optics submodel, which is assigned to the surface energy budget), with the remainder penetrating snow layers absorbed in the top lake body ice layer. +If there are no resolved snow layers, the surface absorption fraction :math:`\beta` is set according to the near-infrared fraction simulated by the atmospheric model. This is apportioned to the surface energy budget (section :numref:`Surface Properties Lake`), and thus no additional radiation is absorbed in the top :math:`z_{a}` (currently 0.6 m) of unfrozen lakes, for which the light extinction coefficient :math:`\eta` (m\ :sup:`-1`) varies between lake columns (equation :eq:`12.41`). For frozen lakes (:math:`T_{g} \le T_{f}` ), the remaining :math:`\left(1-\beta \right)\vec{S}_{g}` fraction of surface absorbed radiation that is not apportioned to the surface energy budget is absorbed in the top lake body layer. This is a simplification, as lake ice is partially transparent. If there are resolved snow layers, then the snow optics submodel (Chapter :numref:`rst_Surface Albedos`) is used to calculate the snow layer absorption (except for the absorption predicted for the top layer by the snow optics submodel, which is assigned to the surface energy budget), with the remainder penetrating snow layers absorbed in the top lake body ice layer. For unfrozen lakes, the solar radiation remaining at depth :math:`z>z_{a}` in the lake body is given by @@ -612,7 +605,7 @@ Finally, the mass of ice in the layer :math:`M_{ice}` is adjusted downwards by : where :math:`c'_{v,i} =c_{v,i} +M\left(c_{liq} -c_{ice} \right)`. -If freezing occurs, :math:`Q_{avail}` is again given by but will be negative. The melt :math:`M`, also negative, is given by +If freezing occurs, :math:`Q_{avail}` is again given by equation :eq:12.47, but will be negative. The melt :math:`M`, also negative, is given by .. math:: :label: 12.51 From 046897f36a4f03ecacf672d861d7b4099eabe5ca Mon Sep 17 00:00:00 2001 From: Naoki Mizukami Date: Wed, 6 May 2026 15:25:15 -0600 Subject: [PATCH 02/11] equation reference syntax fixed --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index cae9fef802..b75ea6f6f4 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -605,7 +605,7 @@ Finally, the mass of ice in the layer :math:`M_{ice}` is adjusted downwards by : where :math:`c'_{v,i} =c_{v,i} +M\left(c_{liq} -c_{ice} \right)`. -If freezing occurs, :math:`Q_{avail}` is again given by equation :eq:12.47, but will be negative. The melt :math:`M`, also negative, is given by +If freezing occurs, :math:`Q_{avail}` is again given by equation :eq:`12.47`, but will be negative. The melt :math:`M`, also negative, is given by .. math:: :label: 12.51 From 594268c1a7391746e4cc360cfa8b9a8dc704b2e6 Mon Sep 17 00:00:00 2001 From: Naoki Mizukami Date: Thu, 28 May 2026 14:24:20 -0600 Subject: [PATCH 03/11] Incorporate suggestions from code review from Keith Adding equation references Co-authored-by: Keith Oleson --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 7fecc55e96..2152d975c0 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -375,7 +375,7 @@ where :math:`\tilde{c}_{v}` is the volumetric heat capacity (J m\ :sup:`-3` K\ : \frac{d}{dt} \sum _{j=1}^{N}\left[\tilde{c}_{v,j} (t)\left(T_{j} -T_{f} \right)+L_{j} (t)\right] \Delta z_{j} =G+\left(1-\beta \right)\vec{S}_{g} -where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Heat Capacities Lake`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing eq. to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in eq.. +where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Heat Capacities Lake`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing equation :eq:`12.27` to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in equation :eq:`12.27' .. _Boundary Conditions Lake: @@ -439,7 +439,7 @@ where N^{2} =\frac{g}{\rho _{i} } \frac{\partial \rho }{\partial z} -and :math:`g` is the acceleration due to gravity (m s\ :sup:`-2`) (:numref:`Table Physical Constants`), :math:`\rho _{i}` is the density of water (kg m\ :sup:`-3`), and :math:`\frac{\partial \rho }{\partial z}` is approximated as :math:`\frac{\rho _{i+1} -\rho _{i} }{z_{i+1} -z_{i} }`. Note that because here, *z* is increasing downwards (unlike in :ref:`Hostetler and Bartlein (1990)`), eq. contains no negative sign; this is a correction from CLM4. The density of water is (:ref:`Hostetler and Bartlein (1990)`) +and :math:`g` is the acceleration due to gravity (m s\ :sup:`-2`) (:numref:`Table Physical Constants`), :math:`\rho _{i}` is the density of water (kg m\ :sup:`-3`), and :math:`\frac{\partial \rho }{\partial z}` is approximated as :math:`\frac{\rho _{i+1} -\rho _{i} }{z_{i+1} -z_{i} }`. Note that because here, *z* is increasing downwards (unlike in :ref:`Hostetler and Bartlein (1990)`), equation :eq:`12.33` contains no negative sign; this is a correction from CLM4. The density of water is (:ref:`Hostetler and Bartlein (1990)`) .. math:: :label: 12.34 @@ -453,7 +453,7 @@ The enhanced diffusivity :math:`K_{ed}` is given by (:ref:`Fang and Stefan 1996< K_{ed} =1.04\times 10^{-8} \left(N^{2} \right)^{-0.43} ,N^{2} \ge 7.5\times 10^{-5} {\rm s}^{2} -where :math:`N^{2}` \ is calculated as in eq. except for the minimum value imposed in. +where :math:`N^{2}` \ is calculated as in equation :eq:`12.33` except for the minimum value imposed in. The thermal conductivity for the liquid water portion of lake body layer *i*, :math:`\tau _{liq,i}` (W m\ :sup:`-1` K\ :sup:`-1`) is given by @@ -612,7 +612,7 @@ If freezing occurs, :math:`Q_{avail}` is again given by equation :eq:`12.47`, bu M=\max \left\{-M_{liq} ,\frac{Q_{avail} }{H_{fus} } \right\} -where :math:`M_{liq}` is the mass of water in the layer: :math:`\left(1-I_{i} \right)\rho _{liq} \Delta z_{i}` for a lake body layer, or the soil / snow water content state variable (:math:`w_{liq}` ). The heat remainder :math:`Q_{rem}` is given by eq. and will be negative or zero. Finally, :math:`M_{liq}` is adjusted downwards by :math:`-M` and the temperature is reset according to eq.. +where :math:`M_{liq}` is the mass of water in the layer: :math:`\left(1-I_{i} \right)\rho _{liq} \Delta z_{i}` for a lake body layer, or the soil / snow water content state variable (:math:`w_{liq}` ). The heat remainder :math:`Q_{rem}` is given by equation :eq:`12.49` and will be negative or zero. Finally, :math:`M_{liq}` is adjusted downwards by :math:`-M` and the temperature is reset according to equation :eq:`12.50` In the presence of nonzero snow water :math:`W_{sno}` without resolved snow layers over an unfrozen top lake layer, the available energy in the top lake layer :math:`\left(T_{1} -T_{f} \right)c_{v,1}` is used to melt the snow. Similar to above, :math:`W_{sno}` is either completely melted and the remainder of heat returned to the top lake layer, or the available heat is exhausted and the top lake layer is set to freezing. The snow thickness is adjusted downwards in proportion to the amount of melt, maintaining constant density. From 3567d7914ff656c9a4ca30f7e802fbd2839d69e5 Mon Sep 17 00:00:00 2001 From: Naoki Mizukami Date: Thu, 28 May 2026 15:30:24 -0600 Subject: [PATCH 04/11] more equation references --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 2152d975c0..fe29f213f6 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -31,7 +31,7 @@ For direct radiation, the albedo *a* for lakes with ground temperature :math:`{T a=\frac{0.5}{\cos z+0.15} -where *z* is the zenith angle. For diffuse radiation, the expression in eq. is integrated over the full sky to yield *a* = 0.10. +where *z* is the zenith angle. For diffuse radiation, the expression in equation :eq:`12.1` is integrated over the full sky to yield *a* = 0.10. For frozen lakes without resolved snow layers, the albedo at cold temperatures *a*\ :sub:`0` is 0.60 for visible and 0.40 for near infrared radiation. As the temperature at the ice surface, :math:`{T}_{g}`, approaches freezing [ :math:`{T}_{f}` (K) (:numref:`Table Physical Constants`)], the albedo is relaxed towards 0.10 based on :ref:`Mironov et al. (2010)`: @@ -293,7 +293,7 @@ The fluxes of momentum, sensible heat, and water vapor are solved for simultaneo #. Heat of vaporization / sublimation :math:`\lambda` \ (equation :eq:`12.8`) -#. Thermal conductivity :math:`\lambda _{T}` \ (above) +#. Thermal conductivity :math:`\lambda _{T}` \ (described above after equation :eq:`12.13`) #. Friction velocity :math:`u_{*}` (equations :eq:`5.32`, :eq:`5.33`, :eq:`5.34`, :eq:`5.35`) @@ -375,7 +375,7 @@ where :math:`\tilde{c}_{v}` is the volumetric heat capacity (J m\ :sup:`-3` K\ : \frac{d}{dt} \sum _{j=1}^{N}\left[\tilde{c}_{v,j} (t)\left(T_{j} -T_{f} \right)+L_{j} (t)\right] \Delta z_{j} =G+\left(1-\beta \right)\vec{S}_{g} -where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Heat Capacities Lake`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing equation :eq:`12.27` to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in equation :eq:`12.27' +where :math:`\tilde{c}_{v,j} (t)`\ is the volumetric heat capacity of the *j*\ th layer (section :numref:`Heat Capacities Lake`), :math:`L_{j} (t)`\ is the latent heat of fusion per unit volume of the *j*\ th layer (proportional to the mass of liquid water present), and the right-hand side represents the net influx of energy to the lake system. Note that :math:`\tilde{c}_{v,j} (t)` can only change due to phase change (except for changing snow layer mass, which, apart from energy required to melt snow, represents an untracked energy flux in the land model, along with advected energy associated with water flows in general), and this is restricted to occur at :math:`T_{j} =T_{f}` \ in the snow-lake-soil system, allowing equation :eq:`12.27` to be precisely enforced and justifying the exclusion of :math:`c_{v,j}` from the time derivative in equation :eq:`12.27`. .. _Boundary Conditions Lake: @@ -423,7 +423,7 @@ The wind-driven eddy diffusion coefficient :math:`\kappa _{e,\, i}` (m\ :sup:`2` \kappa _{e,\, i} =\Bigg\{\begin{array}{rr} {\frac{kw^{*} z_{i} }{P_{0} \left(1+37Ri^{2} \right)} \exp \left(-k^{*} z_{i} \right)\qquad T_{g} >T_{f} } \\ {0\qquad T_{g} \le T_{f} } \end{array} -where :math:`P_{0} =1` is the neutral value of the turbulent Prandtl number, :math:`z_{i}` is the node depth (m), the surface friction velocity (m s\ :sup:`-1`) is :math:`w^{*} =0.0012u_{2}`, and :math:`k^{*}` varies with latitude :math:`\phi` as :math:`k^{*} =6.6u_{2}^{-1.84} \sqrt{\left|\sin \phi \right|}`. For the bottom layer, :math:`\kappa _{e,\, N_{levlak} } =\kappa _{e,N_{levlak} -1\, }`. As in :ref:`Hostetler and Bartlein (1990)`, the 2-m wind speed :math:`u_{2}` (m s\ :sup:`-1`) (eq. ) is used to evaluate :math:`w^{*}` and :math:`k^{*}` rather than the 10-m wind used by :ref:`Henderson-Sellers (1985) `. +where :math:`P_{0} =1` is the neutral value of the turbulent Prandtl number, :math:`z_{i}` is the node depth (m), the surface friction velocity (m s\ :sup:`-1`) is :math:`w^{*} =0.0012u_{2}`, and :math:`k^{*}` varies with latitude :math:`\phi` as :math:`k^{*} =6.6u_{2}^{-1.84} \sqrt{\left|\sin \phi \right|}`. For the bottom layer, :math:`\kappa _{e,\, N_{levlak} } =\kappa _{e,N_{levlak} -1\, }`. As in :ref:`Hostetler and Bartlein (1990)`, the 2-m wind speed :math:`u_{2}` (m s\ :sup:`-1`) (equation :eq:`12.28`) is used to evaluate :math:`w^{*}` and :math:`k^{*}` rather than the 10-m wind used by :ref:`Henderson-Sellers (1985) `. The Richardson number is @@ -453,7 +453,7 @@ The enhanced diffusivity :math:`K_{ed}` is given by (:ref:`Fang and Stefan 1996< K_{ed} =1.04\times 10^{-8} \left(N^{2} \right)^{-0.43} ,N^{2} \ge 7.5\times 10^{-5} {\rm s}^{2} -where :math:`N^{2}` \ is calculated as in equation :eq:`12.33` except for the minimum value imposed in. +where :math:`N^{2}` \ is calculated as in equation :eq:`12.33` except for the minimum value imposed by equation :eq:`12.35`. The thermal conductivity for the liquid water portion of lake body layer *i*, :math:`\tau _{liq,i}` (W m\ :sup:`-1` K\ :sup:`-1`) is given by @@ -552,7 +552,7 @@ where superscripts *n* + 1 and *n* denote values at the end and beginning of the :math:`F_{i}` (W m\ :sup:`-2`) is the downward heat flux at the bottom of layer *i*, and :math:`\phi _{i}` is the solar radiation absorbed in layer *i*. -Eq. is solved using the semi-implicit Crank-Nicholson Method, resulting in a tridiagonal system of equations: +Equation :eq:`12.44` is solved using the semi-implicit Crank-Nicholson Method, resulting in a tridiagonal system of equations: .. math:: :label: 12.45 @@ -697,7 +697,7 @@ Lake Hydrology Overview ^^^^^^^^^^^^^^ -Hydrology is done similarly to other impervious non-vegetated columns (e.g., glaciers) where snow layers may be resolved but infiltration into the permanent ground is not allowed. The water mass of lake columns is currently maintained constant, aside from overlying snow. The water budget is balanced with :math:`q_{rgwl}` (eq.; kg m\ :sup:`-2` s\ :sup:`-1`), a generalized runoff term for impervious land units that may be negative. +Hydrology is done similarly to other impervious non-vegetated columns (e.g., glaciers) where snow layers may be resolved but infiltration into the permanent ground is not allowed. The water mass of lake columns is currently maintained constant, aside from overlying snow. The water budget is balanced with :math:`q_{rgwl}` (kg m\ :sup:`-2` s\ :sup:`-1`), a generalized runoff term for impervious land units that may be negative (see section :numref:`rst_Hydrology` for more information about :math:`q_{rgwl}`). There are some modifications to the soil and snow parameterizations as compared with the soil in vegetated land units, or the snow overlying other impervious columns. The soil can freeze or thaw, with the allowance for frost heave (or the initialization of excess ice) (sections :numref:`Eddy Diffusivity and Thermal Conductivities` and :numref:`Phase Change Lake`), but no air-filled pore space is allowed in the soil. To preserve numerical stability in the lake model (which uses a slightly different surface flux algorithm than over other non-vegetated land units), two changes are made to the snow model. First, dew or frost is not allowed to be absorbed by a top snow layer which has become completely melted during the timestep. Second, because occasional instabilities occurred during model testing when the Courant–Friedrichs–Lewy (CFL) condition was violated, due to the explicit time-stepping integration of the surface flux solution, resolved snow layers must be a minimum of :math:`s_{\min }` = 4 cm thick rather than 1 cm when the default timestep of 1800 s is used. From d54db1dd520a8e3f52eec7f768a87158595f77dc Mon Sep 17 00:00:00 2001 From: Naoki Mizukami Date: Thu, 28 May 2026 15:39:16 -0600 Subject: [PATCH 05/11] fixed numbering bullet points --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index fe29f213f6..74fd10659f 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -318,7 +318,7 @@ The fluxes of momentum, sensible heat, and water vapor are solved for simultaneo where the last term on the right side of equation :eq:`12.23` is the change in saturated specific humidity due to the change in :math:`T_{g}` between iterations. -#. Saturated specific humidity :math:`q_{sat}^{T_{g} }` and its derivative :math:`\frac{dq_{sat}^{T_{g} } }{dT_{g} }` are updated for :math:`T_{g}^{n+1}` (section :numref:`Monin-Obukhov Similarity Theory`). +14. Saturated specific humidity :math:`q_{sat}^{T_{g} }` and its derivative :math:`\frac{dq_{sat}^{T_{g} } }{dT_{g} }` are updated for :math:`T_{g}^{n+1}` (section :numref:`Monin-Obukhov Similarity Theory`). #. Virtual potential temperature scale :math:`\theta _{v*}` (equation :eq:`5.17`) From 16b6cc970876ec7d434f06a74815af4dc5cd2138 Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Tue, 2 Jun 2026 14:43:37 -0700 Subject: [PATCH 06/11] Update doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst Replace reference to the albedo lake section with rst_Surface Albedos --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 74fd10659f..78779cb9a0 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -3,7 +3,7 @@ Lake Model ============= -The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `. Lakes have spatially varying depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`Surface Albedo Lake`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. +The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `. Lakes have spatially varying depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`rst_Surface Albedos`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. .. _Vertical Discretization Lake: From 25feac83d330c0c69d75bf4ae073880669518f86 Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Tue, 2 Jun 2026 14:44:33 -0700 Subject: [PATCH 07/11] Update doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst Replace second reference to albedo lake section with rst_Surface Albedos --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 78779cb9a0..d16c1bee8b 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -226,7 +226,7 @@ The absorbed solar radiation :math:`\vec{S}_{g}` is \vec{S}_{g} =\sum _{\Lambda }S_{atm} \, \downarrow _{\Lambda }^{\mu } \left(1-\alpha _{g,\, \Lambda }^{\mu } \right) +S_{atm} \, \downarrow _{\Lambda } \left(1-\alpha _{g,\, \Lambda } \right) -where :math:`S_{atm} \, \downarrow _{\Lambda }^{\mu }` and :math:`S_{atm} \, \downarrow _{\Lambda }` are the incident direct beam and diffuse solar fluxes (W m\ :sup:`-2`) and :math:`\Lambda` denotes the visible (:math:`<` 0.7\ :math:`\mu {\rm m}`) and near-infrared (:math:`\ge` 0.7\ :math:`\mu {\rm m}`) wavebands (section :numref:`Atmospheric Coupling`), and :math:`\alpha _{g,\, \Lambda }^{\mu }` and :math:`\alpha _{g,\, \mu }` are the direct beam and diffuse lake albedos (section :numref:`Surface Albedo Lake`). +where :math:`S_{atm} \, \downarrow _{\Lambda }^{\mu }` and :math:`S_{atm} \, \downarrow _{\Lambda }` are the incident direct beam and diffuse solar fluxes (W m\ :sup:`-2`) and :math:`\Lambda` denotes the visible (:math:`<` 0.7\ :math:`\mu {\rm m}`) and near-infrared (:math:`\ge` 0.7\ :math:`\mu {\rm m}`) wavebands (section :numref:`Atmospheric Coupling`), and :math:`\alpha _{g,\, \Lambda }^{\mu }` and :math:`\alpha _{g,\, \mu }` are the direct beam and diffuse lake albedos (section :numref:`rst_Surface Albedos`). The net emitted longwave radiation is From 6a5bf980ce6b19d2a05539ec84ae85bd2825e9e7 Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Wed, 3 Jun 2026 11:36:42 -0700 Subject: [PATCH 08/11] Apply suggestion from @slevis-lmwg Removing first half of Surface Albedo section --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 1 - 1 file changed, 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index d16c1bee8b..4fbe0a6551 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -30,7 +30,6 @@ For direct radiation, the albedo *a* for lakes with ground temperature :math:`{T :label: 12.1 a=\frac{0.5}{\cos z+0.15} - where *z* is the zenith angle. For diffuse radiation, the expression in equation :eq:`12.1` is integrated over the full sky to yield *a* = 0.10. For frozen lakes without resolved snow layers, the albedo at cold temperatures *a*\ :sub:`0` is 0.60 for visible and 0.40 for near infrared radiation. As the temperature at the ice surface, :math:`{T}_{g}`, approaches freezing [ :math:`{T}_{f}` (K) (:numref:`Table Physical Constants`)], the albedo is relaxed towards 0.10 based on :ref:`Mironov et al. (2010)`: From 60d5dda8e8bc5f9c5e1702dfe4dba1413f512c2a Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Wed, 3 Jun 2026 12:48:44 -0600 Subject: [PATCH 09/11] Remove Surface Albedo section from Lake Chapter --- .../tech_note/Lake/CLM50_Tech_Note_Lake.rst | 24 ------------------- 1 file changed, 24 deletions(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index 4fbe0a6551..b5806bd322 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -19,30 +19,6 @@ External Data As discussed in :ref:`Subin et al. (2012a, b) `, the Global Lake and Wetland Database (:ref:`Lehner and Doll 2004`) is currently used to prescribe lake fraction in each land model grid cell, for a total of 2.3 million km\ :sup:`-2`. As in :ref:`Subin et al. (2012a, b) `, the :ref:`Kourzeneva et al. (2012)` global gridded dataset is currently used to estimate a mean lake depth in each grid cell, based on interpolated compilations of geographic information. -.. _Surface Albedo Lake: - -Surface Albedo ------------------- - -For direct radiation, the albedo *a* for lakes with ground temperature :math:`{T}_{g}` (K) above freezing is given by (:ref:`Pivovarov, 1972`) - -.. math:: - :label: 12.1 - - a=\frac{0.5}{\cos z+0.15} -where *z* is the zenith angle. For diffuse radiation, the expression in equation :eq:`12.1` is integrated over the full sky to yield *a* = 0.10. - -For frozen lakes without resolved snow layers, the albedo at cold temperatures *a*\ :sub:`0` is 0.60 for visible and 0.40 for near infrared radiation. As the temperature at the ice surface, :math:`{T}_{g}`, approaches freezing [ :math:`{T}_{f}` (K) (:numref:`Table Physical Constants`)], the albedo is relaxed towards 0.10 based on :ref:`Mironov et al. (2010)`: - -.. math:: - :label: 12.2 - - a=a_{0} \left(1-x\right)+0.10x,x=\exp \left(-95\frac{T_{f} -T_{g} }{T_{f} } \right) - -where *a* is restricted to be no less than that given in :eq:`12.1`. - -For frozen lakes with resolved snow layers, the reflectance of the ice surface is fixed at *a*\ :sub:`0`, and the snow reflectance is calculated as over non-vegetated surfaces (Chapter :numref:`rst_Surface Albedos`). These two reflectances are combined to obtain the snow-fraction-weighted albedo as in over non-vegetated surfaces (Chapter :numref:`rst_Surface Albedos`). - .. _Surface Fluxes and Surface Temperature Lake: Surface Fluxes and Surface Temperature From ea613b8ac105d5093b1295aaa340b756654af56b Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Wed, 3 Jun 2026 12:41:46 -0700 Subject: [PATCH 10/11] Apply suggestion from @slevis-lmwg Change section reference from rst_Surface Albedos to Ground Albedos --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index b5806bd322..da6319737e 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -3,7 +3,7 @@ Lake Model ============= -The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `. Lakes have spatially varying depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`rst_Surface Albedos`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. +The lake model, denoted the *Lake, Ice, Snow, and Sediment Simulator* (LISSS), is from :ref:`Subin et al. (2012a) `. It includes extensive modifications to the lake code of :ref:`Zeng et al. (2002) ` used in CLM versions 2 through 4, which utilized concepts from the lake models of :ref:`Bonan (1996) `, :ref:`Henderson-Sellers (1985) `, :ref:`Henderson-Sellers (1986) `, :ref:`Hostetler and Bartlein (1990) `, and the coupled lake-atmosphere model of :ref:`Hostetler et al. (1993) `. Lakes have spatially varying depth prescribed in the surface data (section :ref:`External Data Lake`); the surface data optionally includes lake optical extinction coeffient and horizontal fetch, currently only used for site simulations. Lake physics includes freezing and thawing in the lake body, resolved snow layers, and "soil" and bedrock layers below the lake body. Temperatures and ice fractions are simulated for :math:`N_{levlak} =10` layers (for global simulations) or :math:`N_{levlak} =25` (for site simulations) with discretization described in section :numref:`Vertical Discretization Lake`. Lake albedo is described in section :numref:`Ground Albedos`. Lake surface fluxes (section :numref:`Surface Fluxes and Surface Temperature Lake`) generally follow the formulations for non-vegetated surfaces, including the calculations of aerodynamic resistances (section :numref:`Sensible and Latent Heat Fluxes for Non-Vegetated Surfaces`); however, the lake surface temperature :math:`T_{g}` (representing an infinitesimal interface layer between the top resolved lake layer and the atmosphere) is solved for simultaneously with the surface fluxes. After surface fluxes are evaluated, temperatures are solved simultaneously in the resolved snow layers (if present), the lake body, and the soil and bedrock, using the ground heat flux *G* as a top boundary condition. Snow, soil, and bedrock models generally follow the formulations for non-vegetated surfaces (Chapter :numref:`rst_Soil and Snow Temperatures`), with modifications described below. .. _Vertical Discretization Lake: From 4bb27d6fc39c0a9b510997d77579c715d3518852 Mon Sep 17 00:00:00 2001 From: Samuel Levis Date: Wed, 3 Jun 2026 12:42:14 -0700 Subject: [PATCH 11/11] Apply suggestion from @slevis-lmwg Change section reference from rst_Surface Albedos to Ground Albedos --- doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst index da6319737e..e99fed4c3c 100644 --- a/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst +++ b/doc/source/tech_note/Lake/CLM50_Tech_Note_Lake.rst @@ -201,7 +201,7 @@ The absorbed solar radiation :math:`\vec{S}_{g}` is \vec{S}_{g} =\sum _{\Lambda }S_{atm} \, \downarrow _{\Lambda }^{\mu } \left(1-\alpha _{g,\, \Lambda }^{\mu } \right) +S_{atm} \, \downarrow _{\Lambda } \left(1-\alpha _{g,\, \Lambda } \right) -where :math:`S_{atm} \, \downarrow _{\Lambda }^{\mu }` and :math:`S_{atm} \, \downarrow _{\Lambda }` are the incident direct beam and diffuse solar fluxes (W m\ :sup:`-2`) and :math:`\Lambda` denotes the visible (:math:`<` 0.7\ :math:`\mu {\rm m}`) and near-infrared (:math:`\ge` 0.7\ :math:`\mu {\rm m}`) wavebands (section :numref:`Atmospheric Coupling`), and :math:`\alpha _{g,\, \Lambda }^{\mu }` and :math:`\alpha _{g,\, \mu }` are the direct beam and diffuse lake albedos (section :numref:`rst_Surface Albedos`). +where :math:`S_{atm} \, \downarrow _{\Lambda }^{\mu }` and :math:`S_{atm} \, \downarrow _{\Lambda }` are the incident direct beam and diffuse solar fluxes (W m\ :sup:`-2`) and :math:`\Lambda` denotes the visible (:math:`<` 0.7\ :math:`\mu {\rm m}`) and near-infrared (:math:`\ge` 0.7\ :math:`\mu {\rm m}`) wavebands (section :numref:`Atmospheric Coupling`), and :math:`\alpha _{g,\, \Lambda }^{\mu }` and :math:`\alpha _{g,\, \mu }` are the direct beam and diffuse lake albedos (section :numref:`Ground Albedos`). The net emitted longwave radiation is