richardson_kernel.h

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*/
 
#ifndef RICHARDSON_KERNEL_H
#define RICHARDSON_KERNEL_H
 
#include <cmath>
#include <algorithm>
 
/*
* Similarity laws and corrections for the STABLE regime:
* Based on Mauritsen et al. 2007
*/
 
template<typename T>
__device__ T f_tau_stable(T Ri_b)
{
    return 0.17 * (0.25 + 0.75 / (1.0 + 4.0 * Ri_b));
}
 
template<typename T>
__device__ T f_theta_stable(T Ri_b)
{
    return -0.145 / (1.0 + 4.0 * Ri_b);
}
 
template<typename T>
__device__ T tau_stable(T magu, T Ri_b, T h, T z0, T kappa)
{
    T log_hz0 = log(h / z0);
    return (magu * magu) / (log_hz0 * log_hz0) * (f_tau_stable<T>(Ri_b) / f_tau_stable<T>(0.0)) * (kappa * kappa);
}
 
template<typename T>
__device__ T heat_flux_stable(T ti, T ts, T Ri_b, T h, T z0h, T utau, T kappa, T Pr)
{
    return (ti - ts) / log(h / z0h) * (f_theta_stable<T>(Ri_b) / fabs(f_theta_stable<T>(0.0))) * kappa * (utau / Pr);
}
 
/*
* Similarity laws and corrections for the UNSTABLE (convective) regime:
* Based on Louis 1979
*/
 
template<typename T>
__device__ T f_tau_convective(T Ri_b, T c)
{
    return 1.0 - (2.0 * Ri_b) / (1.0 + c * sqrt(fabs(Ri_b)));
}
 
template<typename T>
__device__ T f_theta_convective(T Ri_b, T c)
{
    // Functionally identical to f_tau_convective in Louis 1979
    return 1.0 - (2.0 * Ri_b) / (1.0 + c * sqrt(fabs(Ri_b)));
}
 
template<typename T>
__device__ T tau_convective(T magu, T Ri_b, T h, T z0, T kappa)
{
    T a = kappa / log(h / z0);
    T b = 2.0;
    T c = 7.4 * (a * a) * b * sqrt(h / z0);
 
    return (a * a) * (magu * magu) * f_tau_convective<T>(Ri_b, c);
}
 
template<typename T>
__device__ T heat_flux_convective(T ti, T ts, T Ri_b, T h, T magu, T z0h, T kappa)
{
    T a = kappa / log(h / z0h);
    T b = 2.0;
    T c = 5.3 * (a * a) * b * sqrt(h / z0h);
 
    return -(a * a) / 0.74 * magu * (ti - ts) * f_theta_convective<T>(Ri_b, c);
}
 
/*
* Similarity laws and corrections for the NEUTRAL regime:
*/
 
template<typename T>
__device__ T tau_neutral(T magu, T h, T z0, T kappa)
{
    T val = (kappa * magu) / log(h / z0);
    return val * val;
}
 
template<typename T>
__device__ T heat_flux_neutral(T ti, T ts, T h, T z0h, T utau, T kappa)
{
    return kappa * utau * (ti - ts) / log(h / z0h);
}
 
/*
 * CUDA kernel for the Richardson wall model.
 */
template<typename T, int BC_TYPE>
__global__ void richardson_compute(
    const T* __restrict__ u_d,
    const T* __restrict__ v_d,
    const T* __restrict__ w_d,
    const T* __restrict__ temp_d,
    const T* __restrict__ temp_w_d,
    const T* __restrict__ h_d,
    const T* __restrict__ n_x_d,
    const T* __restrict__ n_y_d,
    const T* __restrict__ n_z_d,
    T* __restrict__ tau_x_d,
    T* __restrict__ tau_y_d,
    T* __restrict__ tau_z_d,
    int n_nodes,
    T kappa,
    const T * __restrict__ mu_w_d,
    const T * __restrict__ rho_w_d,
    T g1,
    T g2,
    T g3,
    T Pr,
    T z0,
    T z0h_in,
    T bc_value,
    T* __restrict__ Ri_b_diagn,
    T* __restrict__ L_ob_diagn,
    T* __restrict__ utau_diagn,
    T* __restrict__ magu_diagn,
    T* __restrict__ ti_diagn,
    T* __restrict__ ts_diagn,
    T* __restrict__ q_diagn
) {
    const int idx = blockIdx.x * blockDim.x + threadIdx.x;
    const int str = blockDim.x * gridDim.x;
    if(idx >= n_nodes) return;
 
    const T Ri_threshold = 1e-4;
 
    for (int i = idx; i < n_nodes; i += str) {
        T ui = u_d[i];
        T vi = v_d[i];
        T wi = w_d[i];
        T ti = temp_d[i];
        T hi = h_d[i];
        T mu = mu_w_d[i];
        T rho = rho_w_d[i];
 
        // Extract the local normal vector
        T nx = n_x_d[i];
        T ny = n_y_d[i];
        T nz = n_z_d[i];
 
        // Get the tangential component
        T normu = ui * nx + vi * ny + wi * nz;
        ui -= normu * nx;
        vi -= normu * ny;
        wi -= normu * nz;
 
        T magu = sqrt(ui*ui + vi*vi + wi*wi);
        magu = fmax(magu, (T)1e-6);
 
        T utau = kappa * magu / log(hi/z0);
 
        // Zilitinkevich 1995 correlation for thermal roughness
        T z0h;
        if (z0h_in < 0.0) {
            // Note that this uses previous timestep's utau, hence
            // lags behind by one dt. usually very negligible
            z0h = z0 * exp(z0h_in * sqrt((utau*z0)/(mu/rho)));
        } else {
            z0h = z0h_in;
        }
 
        T ts = 0;
        T q  = 0;
 
        // Initialize variables based on Boundary Condition
        if constexpr (BC_TYPE == 0) { // Neumann
            q = bc_value;
        } else {                      // Dirichlet
            ts = bc_value;
            q  = kappa * utau * (ts - ti) / log(hi/z0h);
        }
 
        T Ri_b;
        T g_dot_n = fabs(g1*nx + g2*ny + g3*nz);
 
        // Compute Bulk Richardson Number
        if constexpr (BC_TYPE == 0) {
            Ri_b = -g_dot_n * hi / ti * q / (magu * magu * magu * kappa * kappa);
        } else {
            Ri_b = g_dot_n * hi / ti * (ti - ts) / (magu * magu);
        }
 
        T tau_mag = 0;
 
        // Stability regime branching
        if (Ri_b > Ri_threshold) { // Stable
            tau_mag = tau_stable<T>(magu, Ri_b, hi, z0, kappa);
            utau = sqrt(tau_mag);
            if constexpr (BC_TYPE == 1) {
                q = heat_flux_stable<T>(ti, ts, Ri_b, hi, z0h, utau, kappa, Pr);
            }
        }
        else if (Ri_b < -Ri_threshold) { // Convective
            tau_mag = tau_convective<T>(magu, Ri_b, hi, z0, kappa);
            utau = sqrt(tau_mag);
            if constexpr (BC_TYPE == 1) {
                q = heat_flux_convective<T>(ti, ts, Ri_b, hi, magu, z0h, kappa);
            }
        }
        else { // Neutral
            tau_mag = tau_neutral<T>(magu, hi, z0, kappa);
            utau = sqrt(tau_mag);
            if constexpr (BC_TYPE == 1) {
                q = heat_flux_neutral<T>(ti, ts, hi, z0h, utau, kappa);
            }
        }
 
        // Apply spatial distribution
        tau_x_d[i] = -rho*tau_mag * ui / magu;
        tau_y_d[i] = -rho*tau_mag * vi / magu;
        tau_z_d[i] = -rho*tau_mag * wi / magu;
 
        // Note: L_ob calculation is omitted from the kernel as it is a pure
        // diagnostic variable and writing it to global memory would require
        // passing an extra L_ob_d array pointer if GPU diagnostics are needed.
 
        Ri_b_diagn[i] = Ri_b;
        L_ob_diagn[i] = 9999;
        utau_diagn[i] = utau;
        magu_diagn[i] = magu;
        ti_diagn[i] = ti;
        ts_diagn[i] = temp_w_d[i];
        q_diagn[i] = q;
    }
}
 
#endif // RICHARDSON_KERNEL_H