Ignore the derivative of ∇ᵥp in the ∂u₃ₜ/∂ρ Jacobian block

Author: dennisYatunin

src/prognostic_equations/implicit/manual_sparse_jacobian.jl

@@ -463,15 +463,15 @@ function update_jacobian!(alg::ManualSparseJacobian, cache, Y, p, dtγ, t)
     ∂ᶠu₃_err_∂ᶜρ = matrix[@name(f.u₃), @name(c.ρ)]
     ∂ᶠu₃_err_∂ᶜρe_tot = matrix[@name(f.u₃), @name(c.ρe_tot)]
 
+    ᶜΦ_r = @. lazy(phi_r(thermo_params, ᶜts))
     ᶜθ_v = @. lazy(theta_v(thermo_params, ᶜts))
+    ᶜθ_vr = @. lazy(theta_vr(thermo_params, ᶜts))
     ᶜΠ = @. lazy(exner_function(thermo_params, ᶜts))
     @. ∂ᶠu₃_err_∂ᶜρ =
-        dtγ * (
-            ᶠp_grad_matrix ⋅
-            DiagonalMatrixRow(ᶜkappa_m * (T_0 * cp_d - ᶜK - ᶜΦ)) +
-            DiagonalMatrixRow(cp_d * ᶠinterp(ᶜθ_v) * ᶠgradᵥ(ᶜΠ) / ᶠinterp(ᶜρ)) ⋅
-            ᶠinterp_matrix()
-        )
+        dtγ * DiagonalMatrixRow(
+            (cp_d * ᶠinterp(ᶜθ_v - ᶜθ_vr) * ᶠgradᵥ(ᶜΠ) - ᶠgradᵥ(ᶜΦ_r)) /
+            ᶠinterp(ᶜρ),
+        ) ⋅ ᶠinterp_matrix()
     @. ∂ᶠu₃_err_∂ᶜρe_tot = dtγ * ᶠp_grad_matrix ⋅ DiagonalMatrixRow(ᶜkappa_m)
     ᶜe_tot = @. lazy(specific(Y.c.ρe_tot, Y.c.ρ))
     if MatrixFields.has_field(Y, @name(c.ρq_tot))