Scratch for derived omega in hybrid coordinates

Define p=akP0+bkps(ϕ,λ) p = a_kP_0 + b_k p_s(\phi, \lambda) where we assume that we're working on a single model level.

ω\omega =DpDt=∂p∂t+vh⋅∇ηp+DηDt∂p∂η= \frac{\mathrm{D}p}{\mathrm{D}t} = \frac{\partial p}{\partial t} + \mathbf{v}_h \cdot \nabla_{\eta} p + \frac{\mathrm{D}\eta}{\mathrm{D}t} \frac{\partial p}{\partial \eta}
=0+vh⋅∇ηp−0= 0 + \mathbf{v}_h \cdot \nabla_{\eta} p - 0
=uacos⁡ϕ∂∂ηλp+0⋅∂∂ηϕp= \frac{u}{a\cos\phi} \frac{\partial}{\partial_{\eta} \lambda}p + 0 \cdot \frac{\partial}{\partial_{\eta} \phi}p
=uacos⁡ϕ∂∂ηλp= \frac{u}{a\cos\phi} \frac{\partial}{\partial_{\eta} \lambda}p
=uacos⁡ϕ∂∂ηλ(akp0+bkps(ϕ,λ))= \frac{u}{a\cos\phi} \frac{\partial}{\partial_{\eta} \lambda} \left(a_k p_0 + b_k p_s(\phi, \lambda) \right)
=bkuacos⁡ϕ∂∂ηλ(ps(ϕ,λ))= b_k \frac{u}{a\cos\phi} \frac{\partial}{\partial_{\eta} \lambda} \left(p_s(\phi, \lambda) \right)

Using the definition ps(ϕ,λ)=p0exp⁡[−gRd(τint,1(zs(ϕ,λ))−τint,2(zs(ϕ,λ))IT(ϕ))] p_s(\phi, \lambda) = p_0 \exp \left[ -\frac{g}{R_d}\left(\tau_{\textrm{int}, 1}(z_s(\phi, \lambda)) - \tau_{\textrm{int},2}(z_s(\phi, \lambda))I_T(\phi)\right)\right]

We get that

∂∂ηλps \frac{\partial}{\partial_\eta \lambda} p_s=∂∂ηλp0exp⁡[−gRd(τint,1(zs(ϕ,λ))−τint,2(zs(ϕ,λ))IT(ϕ))]= \frac{\partial}{\partial_\eta \lambda} p_0 \exp \left[ -\frac{g}{R_d}\left(\tau_{\textrm{int}, 1}(z_s(\phi, \lambda)) - \tau_{\textrm{int},2}(z_s(\phi, \lambda))I_T(\phi)\right)\right]
=−gp0Rdexp⁡[−gRd(τint,1(zs(ϕ,λ))−τint,2(zs(ϕ,λ))IT(ϕ))][([∂∂zτint,1](zs(ϕ,λ))−IT(ϕ)[∂∂zτint,2](zs(ϕ,λ)))][∂∂ηλzs](ϕ,λ) = -\frac{gp_0}{R_d} \exp \left[ -\frac{g}{R_d}\left(\tau_{\textrm{int}, 1}(z_s(\phi, \lambda)) - \tau_{\textrm{int},2}(z_s(\phi, \lambda))I_T(\phi)\right)\right] \left[ \left(\left[\frac{\partial}{\partial z}\tau_{\textrm{int}, 1}\right](z_s(\phi, \lambda)) - I_T(\phi) \left[\frac{\partial}{\partial z}\tau_{\textrm{int},2}\right](z_s(\phi, \lambda))\right) \right]\left[\frac{\partial}{\partial_\eta \lambda}z_s\right](\phi, \lambda)
∂zτint,1(z) \partial_z \tau_{\mathrm{int},1}(z) =∂z1Γ[exp⁡(ΓzT0)−1]+z(T0−TPT0TP)exp⁡[−(zgbRdT0)2] = \partial_z\frac{1}{\Gamma} \left[\exp \left(\frac{\Gamma z}{T_0} \right) - 1 \right] + z \left( \frac{T_0 - T_\mathrm{P}}{T_0T_\mathrm{P}} \right) \exp\left[ -\left(\frac{zg}{bR_dT_0} \right)^2\right]
∂zτint,2\partial_z \tau_{\mathrm{int},2} =∂zK+22(TE−TPTETP)z exp⁡[−(zgbRdT0)2] = \partial_z \frac{K+2}{2} \left(\frac{T_\mathrm{E} - T_\mathrm{P}}{T_\mathrm{E} T_\mathrm{P}} \right)z\,\exp \left[-\left(\frac{zg}{bR_dT_0} \right)^2 \right]

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