# Mass-consistent atmospheric energy and moisture budget data from 1979 to present derived from ERA5 reanalysis: Product User Guide

Contributors: Johannes Mayer, Michael Mayer, Leopold Haimberger

# Acronyms

 Acronym Description tediv Divergence of vertical integral of total energy flux tefle Vertical integral of eastward total energy flux tefln Vertical integral of northward total energy flux tetend Tendency of vertical integral of total energy lhdiv Divergence of vertical integral of latent heat flux lhfle Vertical integral of eastward latent heat flux lhfln Vertical integral of northward latent heat flux lhtend Tendency of vertical integral of latent heat wvdiv Divergence of vertical integral of water vapour flux wvfle Vertical integral of eastward water vapour flux wvfln Vertical integral of northward water vapour flux wvtend Tendency of vertical integral of water vapour Re Residual of the dry air mass budget T Temperature in Kelvin Tc Temperature in Celsius q Specific humidity v Horizontal wind field vector pS Surface pressure P Precipitation E Evaporation Φ Geopotential k Kinetic energy of air Lv Latent heat of vaporization g Gravitational acceleration (9.81 m s-2) cp Specific heat capacity of dry air at constant pressure (1004.70 J kg-1 K-1) cv Specific heat capacity of dry air at constant volume (717.65 J kg-1 K-1) cl Specific heat of liquid water (4218.00 J kg-1 K-1) cpv Specific heat of water vapor at constant pressure (1846.10 J kg-1 K-1)

## 1. Introduction

### 1.1. Executive Summary

This dataset provides monthly means of mass-consistent atmospheric energy and moisture budget terms derived from 1-hourly ERA5 reanalysis data. Mass consistency is achieved by iteratively adjusting the wind field every time step. This dataset allows to evaluate atmospheric energy and moisture budget diagnostics for the period from 1979 onward.

### 1.2. Scope of Documentation

This documentation describes the computation of mass-consistent budget terms using 1-hourly analysed state quantities from ERA5.

### 1.3. Version History

No previous versions.

## 2. ProductDescription

### 2.1. Product Overview

#### 2.1.1. Data Description

Table 1: Dataset general attributes

 Dataset attribute Details Data type Gridded Projection Regular grid Horizontal coverage Global Horizontal resolution 0.25° x 0.25° Vertical coverage Surface to top of atmosphere Vertical resolution Single level Temporal coverage 1979/01 - present Temporal resolution Monthly File Format NetCDF 4

Table 2: Variables summary

Table 3: versions history

 Version Release date Changes from previous version 1.0 2022-05-31 (first release)

### 2.2. Input Data

Table 4: Input datasets

 Dataset Summary Variables used ERA5 Provides global 1-hourly analyzed state quantities on 137 atmospheric model levels as well as analyzed surface parameters.  Data are represented either on a reduced Gaussian grid N320 or as spectral coefficients with T639 triangular truncation (see ERA5 data documentation). Temperature, vorticity, divergence, surface geopotential, and logarithm of surface pressure in spherical harmonics. Specific humidity and total column water vapour in grid space.

### 2.3. Method

#### 2.3.1. Background

All ERA5 input fields are transformed (for details see below) to a full Gaussian grid F480 (quadratic grid with respect to the native spectral resolution T639) to avoid aliasing effects. Vorticity and divergence are used to compute the horizontal wind vector at each atmospheric level. Before individual budget terms are computed, the three-dimensional wind field is iteratively adjusted according to the diagnosed imbalance between divergence of vertically integrated dry mass flux and tendency of dry air.  This procedure is repeated every time step.

Mass-consistent wind fields are obtained by computing the residual of the mass continuity of dry air, which reads as follows

$$$$Re = \nabla \cdot \dfrac{1}{g} {\displaystyle\int_0^{p_S}} \left[(1 - q)\textbf{v}\right] \; dp + \dfrac{1}{g} \dfrac{\partial}{\partial t} {\displaystyle\int_0^{p_S}} (1 - q) \; dp,$$$$

where g is the gravitational acceleration, pS is the surface pressure, q is the specific humidity, and v is the horizontal wind vector. The first term on the right side is the divergence of vertically integrated dry mass flux, and second term describes the surface pressure tendency induced by dry air. Inverting the Laplacian of Re and taking the gradient yields a vertically integrated erroneous mass flux, which is converted to a two-dimensional spurious wind field. This spurious divergent wind is subtracted from the original wind field at each level (barotropic wind field correction) making it consistent with the analyzed mass tendency of dry air. After a second iteration of this procedure, mass-adjusted wind fields are used to compute atmospheric energy and moisture budget terms.

##### Atmospheric Energy Budget

Atmospheric energy fluxes and tendencies are computed according to a simplified version of the energy budget as proposed by Mayer et al. (2017), where vertical and lateral enthalpy fluxes associated with water and snow are consistently neglected, such that

$$$$F_{TOA} - \underbrace{\nabla\cdot \dfrac{1}{g} {\displaystyle \int_0^{p_S}} [ (1 - q) c_p T_c + L_v (T_c) q + \Phi + k] \; \textbf{v}\; dp}_{\text{tediv} = \nabla\cdot (\text{tefle, tefln})^T} - \underbrace{\dfrac{\partial}{\partial t} \dfrac{1}{g} {\displaystyle\int_0^{p_S}} [ (1 - q) c_v T_c + L_v (T_c) q + \Phi + k] dp}_{\text{tetend}} - F_S = 0$$$$

where cp is the specific heat capacity of dry air at constant pressure, Tc is the air temperature measured in Celsius, Lv(T) is the temperature-dependent latent heat of vaporization, Φ is the potential energy, k is the kinetic energy, and cv is the specific heat capacity of dry air at constant volume. The vertical fluxes FTOA and FS describe the net energy flux at the top of the atmosphere and net surface heat (radiative plus turbulent) flux, which are both not included in this dataset. The second term describes the divergence of the vertical integral of moist static plus kinetic energy flux (i.e., the divergence of north- and eastward energy fluxes), and the third term is the tendency of the vertical integral of total energy. Note that the total energy is the sum of internal energy cvTc, latent heat Lvq, potential energy Φ, and kinetic energy k, whereas the moist static plus kinetic energy contains the sensible heat cpTc instead of the internal energy. For the sake of simplicity, however, the 'moist static plus kinetic energy' as used in the divergence term is also referred to as the 'total energy' in this dataset, although it contains the sensible heat and not the internal energy. The temperature-dependent latent heat of vaporization is computed according to the IFS documentation, Part IV, and is defined as

$$L_v(T_c) = L_v(T_0) + (c_{pv} - c_l)*(T - T_0),$$

where Lv(T0) = 2.5008x106 J kg-1  is the latent heat at the triple point temperature T0 (in Kelvin), cpv is the specific heat capacity of water vapour at constant pressure, cl is the specific heat of liquid water, and T is the air temperature measured in Kelvin. To derive energy budget terms with constant latent heat of vaporization (as provided by ERA5), latent heat terms can be subtracted and replaced by corresponding water vapour terms multiplied by  Lv(T0).  The potential energy Φ is computed as described in the IFS documentation, Part III.

##### Atmospheric Moisture Budget

The atmospheric moisture budget can be written as

$$\underbrace{\nabla \cdot \dfrac{1}{g} {\displaystyle\int_0^{p_S}} (q \textbf{v}) \; dp}_{\text{wvdiv} = \nabla\cdot (\text{wvfle, wvfln})^T} + \underbrace{\dfrac{1}{g} \dfrac{\partial}{\partial t} {\displaystyle\int_0^{p_S}} q \; dp}_{\text{wvtend}} + P + E = 0,$$

where precipitation P and evaporation E (not in this dataset) are surface mass fluxes in units kg m-2 s-1.  The first term describes the divergence of the vertical integral of atmospheric water vapour flux, the second term describes the tendency of the vertical integral of atmospheric water vapour (i.e., total column vapour). That is, atmospheric fluxes and tendencies of water vapour must balance surface freshwater fluxes P+E. The divergence term of the moisture budget also employs mass-adjusted wind fields v, albeit it is affected only weakly by spurious divergent winds. Note that tendency terms in this dataset are computed as exact difference from 00 UTC at the first of month to 00 UTC at the first of following month divided by the number of seconds.

#### 2.3.2. Model / Algorithm

The following pseudo code describes the mass-adjustment procedure and subsequent computation of energy and moisture budget terms. All spectral transformations (i.e., gradient and divergence computations, Laplace inversion) were performed with routines from OpenIFS .

\begin{align} &\text{for each time step do } \\ &\qquad \Phi_S \;\;\; \leftarrow \text{ Read surface geopotential} \\ &\qquad vort \: \leftarrow \text{ Read vorticity} \\ &\qquad div \;\;\; \leftarrow \text{ Read divergence} \\ &\qquad T \;\;\;\;\;\;\; \leftarrow \text{ Read temperature} \\ &\qquad q \;\;\;\;\;\;\;\: \leftarrow \text{ Read specific humidity} \\ &\qquad p_S \;\;\;\;\: \leftarrow \text{ Read logarithm of surface pressure} \\ &\qquad tcwv \leftarrow \text{ Read total column water vapour} \\ &\qquad \\ &\qquad \text{Transform all input fields to full Gaussian grid F480} \\ &\qquad \\ &\qquad \textbf{v} \leftarrow \text{ Compute horizontal wind field using } vort, div \\ &\qquad wvtend \leftarrow \text{ Compute tendency of the vertical integral of water vapour using } tcwv \\ &\qquad mtend \;\;\leftarrow \text{ Compute tendency of vertically integrated atmospheric mass using } p_S \\ &\qquad \\ &\qquad \text{for each correction step do } \\ &\qquad\qquad mdiv \;\;\;\leftarrow \text{ Compute divergence of vertically integrated atmospheric mass flux using } \textbf{v} \\ &\qquad\qquad wvdiv \;\leftarrow \text{ Compute vertically integrated water vapour divergence using } \textbf{v}, q \\ &\qquad\qquad errdiv \leftarrow mdiv - wvdiv + mtend - wvtend \\ &\qquad\qquad \textbf{v}_{err} \;\;\;\;\;\; \leftarrow \text{ Compute spurious two-dimensional wind field using } errdiv \\ &\qquad\qquad \text{ for each atmospheric level } i \text{ in } \textbf{v} \text{ do } \textbf{v}_i \leftarrow \textbf{v}_i - \textbf{v}_{err} \\ &\qquad \text{end do} \\ &\qquad \\ &\qquad T_c \leftarrow T - 273.15 \\ &\qquad lhtend \;\;\leftarrow \text{ Compute tendency of the vertical integral of latent heat using } q, T_c \\ &\qquad tetend \;\;\leftarrow \text{ Compute tendency of the vertical integral of total energy using } \textbf{v}, q, T_c, \Phi_S \\ &\qquad lhfle, lhfln \;\;\;\leftarrow \text{ Compute vertical integral of latent heat fluxes using } \textbf{v}, q, T_c \\ &\qquad tefle, tefln \;\;\;\;\leftarrow \text{ Compute vertical integral of total energy fluxes using } \textbf{v}, q, T_c, \Phi_S \\ &\qquad wvfle, wvfln \leftarrow \text{ Compute vertical integral of water vapour fluxes using } \textbf{v}, q \\ &\qquad lhdiv \;\;\leftarrow \text{ Compute divergence of the vertical integral of latent heat fluxes using } lhfle, lhfln \\ &\qquad tediv \;\;\leftarrow \text{ Compute divergence of the vertical integral of total energy fluxes using } tefle, tefln \\ &\qquad wvdiv \leftarrow \text{ Compute divergence of the vertical integral of water vapour fluxes using } wvfle, wvfln \\ &\text{end do} \end{align}

#### 2.3.3. Validation

The divergence fields in this dataset exhibit zero global mean suggesting  optimal computations and good accuracy. Tendency terms are temporally stable and exhibit long-term global zero mean indicating good reliability.  Indirectly estimated oceanic FS derived from tediv and tetend in combination with FTOA from CERES-EBAF (not in this dataset) agrees with the observation-based ocean heat uptake to within 1 W m-2 (see Mayer et al. 2022).  All fields are in good qualitative agreement with known patterns of the respective quantities, but satisfaction of physical constraints (e.g., magnitude of ocean-to-land energy and moisture transport or temporal stability) is much improved compared to earlier evaluations (see Mayer et al. 2021 and 2022 for comprehensive evaluation).

## 3. Known issues

1. The divergence terms (tediv, lhdiv, wvdiv) with full spectral resolution show artificial pattern of numerical noise over high topography, which are thus spectrally truncated at wave number 180. The divergence fields with full spectral resolution (see example in Fig. 1) can be reconstructed by computing the divergence of corresponding north- and eastward fluxes provided in this dataset.
2. The ocean-to-land energy transport as estimated from tediv exhibits an unrealistically strong gradual change in the late 1990s and early 2000s, which likely stems from changes in the observing system that has been assimilated by ERA5 (see Mayer et al. 2021 for discussion).
3. Global ocean and land averages of wvdiv exhibit a reasonably strong but statistically insignificant trend over the available period, see Mayer et al. (2021) for further details.

Figure 1: The divergence of the vertical integral of total energy flux (left) truncated at wave number 180, and (right) with full spectral resolution T639.

## 4. Licence, Acknowledgement and Citation

This dataset is provided under the licence to use Copernicus Products.

All users of this dataset must:

• acknowledge according to the licence to use Copernicus Products
• provide clear and visible attribution to the Copernicus programme by citing the web Climate Data Store (CDS) catalogue entry as follows:

Mayer, J., Mayer, M., Haimberger, L.,(2022): Mass-consistent atmospheric energy and moisture budget data from 1979 to present derived from ERA5 reanalysis, v1.0, Copernicus Climate Change Service (C3S) Climate Data Store (CDS). (Accessed on 31-05-2022), https://doi.org/10.24381/cds.c2451f6b.

Please refer to How to acknowledge and cite a Climate Data Store (CDS) catalogue entry and the data published as part of it for complete details.

The authors of this dataset are financially supported by the Austrian Science Funds project P33177. The dataset is created as in-kind contribution to Copernicus.

# References

Mayer, J., Mayer, M. and Haimberger, L., (2022). Comparison of Surface Energy Fluxes from Global to Local Scale. Accepted in Journal of Climate. https://doi.org/10.1175/JCLI-D-21-0598.1

Mayer, J., Mayer, M. and Haimberger, L., (2021). Consistency and Homogeneity of Atmospheric Energy, Moisture, and Mass Budgets in ERA5. Journal of Climate 34(10), 3955-3974. https://doi.org/10.1175/JCLI-D-20-0676.1

Mayer, M., Haimberger, L., Edwards, J. M., and Hyder, P. (2017). Toward consistent diagnostics of the coupled atmosphere and ocean energy budgets. Journal of Climate, 30(22), 9225-9246. https://doi.org/10.1175/JCLI-D-17-0137.1

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