Simulation of very severe cyclone Mala over Bay of Bengal with HWRF modeling system

Abstract

Tropical cyclone is one of the most devastating weather phenomena all over the world. The Environmental Modeling Center (EMC) of the National Center for Environmental Prediction (NCEP) has developed a sophisticated mesoscale model known as Hurricane Weather Research and Forecasting (HWRF) system for tropical cyclone studies. The state-of-the-art HWRF model (atmospheric component) has been used in simulating most of the features our present study of a very severe tropical cyclone “Mala”, which developed on April 26 over the Bay of Bengal and crossed the Arakan coast of Myanmar on April 29, 2006. The initial and lateral boundary conditions are obtained from Global Forecast System (GFS) analysis and forecast fields of the NCEP, respectively. The performance of the model is evaluated with simulation of cyclone Mala with six different initial conditions at an interval of 12 h each from 00 UTC 25 April 2006 to 12 UTC 27 April 2006. The best result in terms of track and intensity forecast as obtained from different initial conditions is further investigated for large-scale fields and structure of the cyclone. For this purpose, a number of important predicted fields’ viz. central pressure/pressure drop, winds, precipitation, etc. are verified against observations/verification analysis. Also, some of the simulated diagnostic fields such as relative vorticity, pressure vertical velocity, heat fluxes, precipitation rate, and moisture convergences are investigated for understanding of the characteristics of the cyclone in more detail. The vector displacement errors in track forecasts are calculated with the estimated best track provided by the India Meteorological Department (IMD). The results indicate that the model is able to capture most of the features of cyclone Mala with reasonable accuracy.

Appendix 1

1.1 HWRF modeling system

HWRF is a fully compressible Eulerian non-hydrostatic coordinate system with a hydrostatic option. The horizontal rotated latitude-longitude coordinate and the vertical terrain following hydrostatic-pressure-based sigma coordinate system is used. The sigma coordinate system is up to a specified pressure surface usually below the tropopause, and hydrostatic pressure coordinate above. Top of the model is a constant pressure surface.

1.2 Model formulation

The vertical sigma coordinate is defined as,

$$ \begin{aligned} \sigma =\, & \frac{{\pi - \pi_{t} }}{\mu } \\ \mu =\, & \pi_{s} - \pi_{t} \\ \end{aligned} $$

where π is the hydrostatic pressure, π _s stands for the hydrostatic pressure at the surface, π _t represents the hydrostatic pressure at the top of the model atmosphere.

Then the equations governing a dry, inviscid and adiabatic non-hydrostatic atmosphere are (Janjic et al. 2001),

$$ \frac{{\partial {\mathbf{v}}}}{\partial t} = - {\mathbf{v}} \cdot \nabla_{\sigma } {\mathbf{v}} - \dot{\sigma }\frac{{\partial {\mathbf{v}}}}{\partial \sigma } - (1 + \varepsilon )\nabla_{\sigma } \Upphi - \alpha \nabla_{\sigma } p + f{\mathbf{k}} \times {\mathbf{v}}\quad \left( {\text{Horizontal momentum}} \right) $$

$$ \frac{\partial T}{\partial t} = - {\mathbf{v}} \cdot \nabla_{\sigma } T - \dot{\sigma }\frac{\partial T}{\partial \sigma } + \frac{\alpha }{{C_{p} }}\left[ {\frac{\partial p}{\partial t} + {\mathbf{v}} \cdot \nabla_{\sigma } p + \dot{\sigma }\frac{\partial p}{\partial \sigma }} \right]\quad \left( {\text{Thermodynamic}} \right) $$

$$ \frac{\partial \mu }{\partial t} + \nabla_{\sigma } \cdot (\mu {\mathbf{v}}) + \frac{{\partial (\mu \dot{\sigma })}}{\partial \sigma } = 0\quad \left( {\text{Continuity}} \right) $$

$$ \frac{\partial p}{\partial \pi } = 1 + \varepsilon \quad \left( {\text{Vertical equation of motion}} \right) $$

$$ \frac{\partial \Upphi }{\partial \sigma } = - \mu \frac{RT}{p}\quad ({\text{Hypsometric}}) $$

$$ w = \frac{1}{g}\frac{{{\text{d}}\Upphi }}{{{\text{d}}t}} = \frac{1}{g}\left( {\frac{\partial \Upphi }{\partial t} + {\mathbf{v}} \cdot \nabla_{\sigma } \Upphi + \dot{\sigma }\frac{\partial \Upphi }{\partial \sigma }} \right)\quad ({\text{Non-hydrostatic continuity}}) $$

$$ \alpha = {\raise0.7ex\hbox{${RT}$} \!\mathord{\left/ {\vphantom {{RT} p}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$p$}}\quad ({\text{Equation of state}}) $$

$$ \varepsilon = \frac{1}{g}\frac{{{\text{d}}w}}{{{\text{d}}t}}\quad ({\text{Definition of }}\varepsilon ) $$

where v is the horizontal wind vector, p is the non-hydrostatic pressure, R is the gas constant for dry air, T is temperature and Φ is geopotential.

It is to be noted that, Φ, w, and ε are not independent and there is no independent prognostic equations for w. Instead, w is computed diagnostically from the non-hydrostatic continuity equation. The parameter ε is the central point of attention for non-hydrostatic dynamics. In meso- and large-scale atmospheric flows, ε << 1.

1.3 Discretization of equations

The discretization of model equations is as follows.

1.4 Horizontal discretization equation

In order to organize the spurious accumulation of energy at the smallest resolvable scales, an energy and enstrophy conserving advection scheme for the semi-staggered E-grid is used (Arakawa and Lamb 1977).

The horizontal advection scheme for the E-grid is defined by,

$$ \begin{aligned} U = & \overline{\Updelta \pi }^{\lambda } u2\Updelta x \\ V = & \overline{\Updelta \pi }^{\varphi } v2\Updelta y \\ U^{\prime } = & \overline{\Updelta \pi }^{{\lambda^{\prime}}} \left( {\overline{u\Updelta y + v\Updelta x}^{\varphi \prime } } \right) \\ V^{\prime } = & \overline{\Updelta \pi }^{\varphi \prime } \left( {\overline{ - u\Updelta y + v\Updelta x}^{{\lambda^{\prime}}} } \right) \\ A = & 4\Updelta x\Updelta y,A^{\prime } = 2\Updelta x\Updelta y \\ \end{aligned} $$

where Δπ is the thickness of the layer in hydrostatic pressure and $\Updelta x = a\cos (\varphi )\Updelta \lambda ,\Updelta y = a\Updelta \varphi .$ Then the horizontal advection for T, u and v are defined as,

$$ \begin{aligned} \frac{\partial T}{\partial t} = & - \frac{1}{\Updelta \pi }\left[ {\frac{1}{3}\frac{1}{A}\left( {\overline{{U\Updelta_{\lambda } T}}^{\lambda } + \overline{{V\Updelta_{\varphi } T}}^{\varphi } } \right) + \frac{2}{3}\frac{1}{{A^{\prime } }}\left( {\overline{{U^{\prime } \Updelta_{{\lambda^{\prime } }} T}}^{{\lambda^{\prime } }} + V\Updelta_{\varphi } T^{{\varphi^{\prime } }} } \right)} \right] + \cdots \\ \frac{\partial u}{\partial t} = & - \frac{1}{{\overline{\Updelta \pi }^{\lambda } }}\left[ {\frac{1}{3}\frac{1}{A}\left( {\overline{{\overline{U}^{\lambda } \Updelta_{\lambda } u}}^{\lambda } + \overline{{\overline{V}^{\lambda } \Updelta_{\varphi } u}}^{\varphi } } \right) + \frac{2}{3}\frac{1}{{A^{\prime } }}\left( {\overline{{\overline{{U^{\prime } }}^{\lambda } \Updelta_{{\lambda^{\prime } }} u}}^{{\lambda^{\prime } }} + \overline{{\overline{{V^{\prime } }}^{\lambda } \Updelta_{{\varphi^{\prime } }} u}}^{{\varphi^{\prime } }} } \right)} \right] + \cdots \\ \frac{\partial v}{\partial t} = & - \frac{1}{{\overline{\Updelta \pi }^{\varphi } }}\left[ {\frac{1}{3}\frac{1}{A}\left( {\overline{{\overline{U}^{\varphi } \Updelta_{\lambda } v}}^{\lambda } + \overline{{\overline{V}^{\varphi } \Updelta_{\varphi } v}}^{\varphi } } \right) + \frac{2}{3}\frac{1}{{A^{\prime } }}\left( {\overline{{\overline{{U^{\prime } }}^{\varphi } \Updelta_{{\lambda^{\prime } }} v}}^{{\lambda^{\prime } }} + \overline{{\overline{{V^{\prime } }}^{\varphi } \Updelta_{{\varphi^{\prime } }} v}}^{{\varphi^{\prime } }} } \right)} \right] + \cdots \\ \end{aligned} $$

In horizontal co-ordinate system, in order to achieve higher computational efficiency of the model code, a transformed latitude-longitude coordinate system is used. This system is obtained by rotation of the natural geodesic latitude-longitude in such a way that the intersection of the equator and zero meridian of the transformed system (λ ₀, φ₀) coincides with the center of the model domain. The transformation and inverse transformation equations between the rotated and the natural latitude-longitude system (λ, φ) are as follows .

$$ \begin{aligned} \Uplambda = & \arctan \frac{{\cos \varphi \sin (\lambda - \lambda_{0} )}}{{\cos \varphi_{0} \cos \varphi \cos (\lambda - \lambda_{0} ) + \sin \varphi_{0} \sin \varphi }} \\ \Upphi = & \arcsin (\cos \varphi_{0} \sin \varphi - \sin \varphi_{0} \cos \varphi \cos (\lambda - \lambda_{0} )) \\ \varphi = & \arcsin (\sin \varphi_{0} \cos \Upphi \cos \Uplambda + \cos \varphi_{0} \sin \Upphi \\ \lambda =\, & \lambda_{0} + \arcsin \left( {\frac{\sin \Uplambda \cos \Upphi }{{\cos \varphi_{0} }}} \right) \\ \end{aligned} $$

The horizontal wind in the rotated system V = (U, V) expressed in terms of the wind in the natural latitude/longitude system v = (u,v) is defined as;

$$ \begin{aligned} U = & \frac{{u\left[ {\cos \varphi_{0} \cos \varphi + \sin \varphi_{0} \sin \varphi \cos (\lambda - \lambda_{0} )} \right] - v\sin \varphi_{0} \sin (\lambda - \lambda_{0} )}}{{\sqrt {1 - \left[ {\cos \varphi_{0} \sin \varphi - \sin \varphi_{0} \cos \varphi \cos (\lambda - \lambda_{0} )} \right]^{2} } }} \\ V = & \frac{{u\sin \varphi_{0} \sin (\lambda - \lambda_{0} ) + v\left[ {\cos \varphi_{0} \cos \varphi + \sin \varphi_{0} \sin \varphi \cos (\lambda - \lambda_{0} )} \right]}}{{\sqrt {1 - \left[ {\cos \varphi_{0} \sin \varphi - \sin \varphi_{0} \cos \varphi \cos (\lambda - \lambda_{0} )} \right]} }} \\ \end{aligned} $$

The inverse wind transformation is given by,

$$ \begin{aligned} u = & \frac{{U(\cos \varphi_{0} \cos \Upphi - \sin \varphi_{0} \sin \Upphi \cos \Uplambda ) + V\sin \varphi_{0} \sin \Uplambda }}{{\sqrt {1 - (\cos \varphi_{0} \sin \Upphi + \sin \varphi_{0} \cos \Uplambda \cos \Upphi )^{2} } }} \\ v = & \frac{{ - U\sin \varphi_{0} \sin \Uplambda + V(\cos \varphi_{0} \cos \Upphi - \sin \varphi_{0} \sin \Upphi \cos \Uplambda }}{{\sqrt {1 - (\cos \varphi_{0} \sin \Upphi + \sin \varphi_{0} \cos \Uplambda \cos \Upphi )^{2} } }} \\ \end{aligned} $$

1.5 Vertical discretization equations

The Lorenz vertical staggering of the variables is used in the vertical (Janjic 1977). The geopotential and non-hydrostatic pressure are defined at the interfaces of the layers, while all three velocity components and temperature are carried in the middle layers of the model. The vertical velocity is defined at the E-grid mass points.

The terrain following hydrostatic-pressure-based sigma coordinate up to a specified pressure surface usually below the tropopause, and hydrostatic pressure coordinate above. Top of the model is constant pressure surface.

In the pressure range,

$$ \nabla_{p} \cdot ({\mathbf{v}}) + \frac{\partial \omega }{\partial p} = 0 $$

In the sigma range,

$$ \frac{\partial PD}{\partial t} + \nabla_{\sigma } \cdot (PD\,{\mathbf{v}}) + \frac{{\partial (PD\dot{\sigma })}}{\partial \sigma } = 0 $$

where as $PD\dot{\sigma } = \omega$ at the vertical level of the model where sigma range changes to the pressure range.

Hence, the mass continuity equation can be written as:

For the pressure range:

$$ 0 = - \left[ {\frac{1}{3}\frac{1}{A}\left( {\Updelta_{\lambda } U + \Updelta {}_{\varphi }V} \right) + \frac{2}{3}\frac{1}{{A^{\prime } }}\left( {\Updelta_{{\lambda^{\prime } }} U^{\prime } + \Updelta_{{\varphi^{\prime } }} V^{\prime } } \right)} \right] - \Updelta_{\pi } \omega $$

For the sigma range:

$$ \frac{\partial \Updelta \pi }{\partial t} = - \left[ {\frac{1}{3}\frac{1}{A}\left( {\Updelta_{\lambda } U + \Updelta {}_{\varphi }V} \right) + \frac{2}{3}\frac{1}{{A^{\prime } }}\left( {\Updelta_{{\lambda^{\prime } }} U^{\prime } + \Updelta_{{\varphi^{\prime } }} V^{\prime } } \right)} \right] - \Updelta_{\sigma } \left( {\mu \dot{\sigma }} \right) $$

For the interface between sigma and pressure range:

$$ (\mu \dot{\sigma })_{I} = \omega_{I} $$

1.6 Time discretization equations

HWRF uses four types of time integration: (i) modified Adams–Bashforth scheme for horizontal advection of u, v and T, and Coriolis terms, (ii) Crank–Nicholson for vertical advection of u, v and T, (iii) forward–backward scheme for adjustment terms and (iv) an implicit scheme for vertically propagating sound waves.

The Adam-Bashforth scheme can be represented as:

$$ \frac{{y^{\tau + 1} - y^{\tau } }}{\Updelta t} = \frac{3}{2}f\left( {y^{\tau } } \right) - \frac{1}{2}f\left( {y^{\tau - 1} } \right) $$

This method has a slight linear instability, which can be stabilized as;

$$ \frac{{y^{\tau + 1} - y^{\tau } }}{\Updelta t} = 1.533f\left( {y^{\tau } } \right) - 0.533f\left( {y^{\tau - 1} } \right) $$

The Crank–Nicholson scheme can be represented as:

$$ \frac{{y^{\tau + 1} - y^{\tau } }}{\Updelta t} = \frac{1}{2}\left[ {f\left( {y^{\tau + 1} } \right) + f\left( {y^{\tau } } \right)} \right] $$

This is an implicit scheme and so is always stable.

The time integration for the terms involving the propagation of gravity waves is handled by a forward–backward process.

Using the shallow water equation,

$$ \frac{\partial u}{\partial t} = - g\frac{\partial h}{\partial x},\quad \frac{\partial h}{\partial t} = - H\frac{\partial u}{\partial x} $$

The mass tendency equation is advanced by a forward step:

$$ h^{\tau + 1} = h^{\tau } - \Updelta t\;H\;\frac{{\partial u^{\tau } }}{\partial x} $$

The velocity equations are then advanced with a backward step:

$$ u^{\tau + 1} = u^{\tau } - \Updelta t\;g\;\frac{{\partial h^{\tau + 1} }}{\partial x} $$

The time integration for vertically propagating sound waves is:

$$ \frac{{\partial^{2} p^{\prime } }}{{\partial t^{2} }} = \frac{{p^{\prime n + 1} - 2p^{\prime n} + p^{\prime n - 1} }}{{\Updelta t^{2} }} = \frac{{{\mathbf{c}}_{p} }}{{{\mathbf{c}}_{v} }}R\,T_{0} \;\frac{{\partial^{2} p^{\prime n + 1} }}{{\partial z_{0}^{2} }}, $$

where p′ represents a perturbation pressure

The advection of other variables such as water vapor and liquid water uses the Langangian forward time differencing, which can be represented as:

$$ \frac{{y_{j}^{\tau + 1 * } - y_{j}^{\tau } }}{\Updelta t} = - u\frac{{y_{j}^{\tau } - y_{j - 1}^{\tau } }}{\Updelta x},\quad u > 0 $$

This is followed by a negative diffusion step to reduce smoothing:

$$ \frac{{y_{j}^{\tau + 1} - y_{j}^{\tau + 1 * } }}{\Updelta t} = - f\left( {\frac{u\Updelta t}{\Updelta x}} \right)\frac{{y_{j - 1}^{\tau + 1 * } - 2y_{j}^{\tau + 1 * } + y_{j + 1}^{\tau + 1 * } }}{\Updelta x},\quad f\left( {\frac{u\Updelta t}{\Updelta x}} \right) > 0 $$

With the non-hydrostatic approach, with vanishing ɛ, the time discrete equation with adding Crank–Nicholson vertical and Adams–Bashforth horizontal advection can be written as:

$$ T_{1}^{ * } = T^{n} - \Updelta t\left( {{\mathbf{v}}^{n} \cdot \nabla_{\sigma } T^{n + 1/2} + \dot{\sigma }^{n} \frac{{\partial T^{n + 1/2} }}{\partial \sigma }} \right) $$

Finally

$$ T_{1} = T_{1}^{ * } + \frac{\Updelta t}{{{\mathbf{c}}_{p} }}\,\frac{{RT^{n} }}{{p^{n} }}\omega_{1} $$

1.7 Lateral boundary condition

The velocity and mass variables are specified only on the outermost rows and columns. On the first row or column from the outer boundaries, each variable is replaced by a four point average from its surrounding points (Mesinger and Janjic 1974, Mesinger 1977). Inside the domain, the initial values before the averaging are defined by the tendency equations with the advection terms computed using upstream differences on the three rows next to the internal boundary where the four-point averaging is applied. The enhanced divergence damping is generally applied close to the boundaries.

The vertical boundary conditions are:

$$ \dot{\sigma } = 0\quad {\text{and}}\quad p - \pi = 0\;{\text{at}}\;\sigma = 0 $$

and

$$ \dot{\sigma } = 0\;\quad {\text{and}}\quad \;\frac{\partial (p - \pi )}{\partial \sigma } = 0\;{\text{at}}\;\sigma = 1 $$