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<p>Anthropogenic aerosols impact cirrus clouds through ice nucleation, thereby changing the Earth's radiation budget. However, the magnitude and sign of anthropogenic forcing in cirrus clouds is still very uncertain depending on the treatments for ice-nucleating particles (INPs), the treatments for haze particle freezing, and the ice nucleation scheme. In this study, a new ice nucleation scheme (hereafter the HYBRID scheme) is developed to combine the best features of two previous ice nucleation schemes, so that global models are able to calculate the ice number concentration in both updrafts and downdrafts associated with gravity waves, and it has a robust sensitivity to the change of aerosol number. The scheme is applied in a box model, and the ice number concentrations (<span class="inline-formula">9.52±2.08</span>&thinsp;L<span class="inline-formula">⁻¹</span>) are somewhat overestimated but are in reasonable agreement with those from an adiabatic parcel model (<span class="inline-formula">9.40±2.31</span>&thinsp;L<span class="inline-formula">⁻¹</span>). Then, the forcing and cloud changes associated with changes in aircraft soot, sulfur emission, and all anthropogenic emissions between the preindustrial (PI) period and the present day (PD) are examined using the CESM/IMPACT global model with the HYBRID scheme. Aircraft soot emissions decrease the global average ice number concentration (<span class="inline-formula"><i>N</i>_i</span>) by <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M6" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">1.0</mn><mo>±</mo><mn mathvariant="normal">2.4</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">7</mn></msup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="79pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="5e9b9a236dabf072177e8fd2e3451cb5"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00001.svg" width="79pt" height="13pt" src="acp-20-7801-2020-ie00001.png"/></svg:svg></span></span>&thinsp;m<span class="inline-formula">⁻²</span> (<span class="inline-formula">−1</span>&thinsp;%) (over the entire column) due to the inhibition of homogeneous nucleation and lead to a radiative forcing of <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M9" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">0.14</mn><mo>±</mo><mn mathvariant="normal">0.07</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="64pt" height="10pt" class="svg-formula" dspmath="mathimg" md5hash="d53f9c265767715b6df4cc07da936ba3"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00002.svg" width="64pt" height="10pt" src="acp-20-7801-2020-ie00002.png"/></svg:svg></span></span>&thinsp;W&thinsp;m<span class="inline-formula">⁻²</span>, while the increase in sulfur emissions increases the global average <span class="inline-formula"><i>N</i>_i</span> by <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M12" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">7.3</mn><mo>±</mo><mn mathvariant="normal">2.9</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">7</mn></msup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="72pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="d3d5761f7954999318e81e73b29653dd"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00003.svg" width="72pt" height="13pt" src="acp-20-7801-2020-ie00003.png"/></svg:svg></span></span>&thinsp;m<span class="inline-formula">⁻²</span> (5&thinsp;%) due to the increase in homogeneous nucleation and leads to a radiative forcing of <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M14" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">0.02</mn><mo>±</mo><mn mathvariant="normal">0.06</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="64pt" height="10pt" class="svg-formula" dspmath="mathimg" md5hash="708d1a3803d2349359bd3c5dff6ae45a"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00004.svg" width="64pt" height="10pt" src="acp-20-7801-2020-ie00004.png"/></svg:svg></span></span>&thinsp;W&thinsp;m<span class="inline-formula">⁻²</span>. The possible effects of aerosol and cloud feedbacks to the meteorological state in remote regions partly contribute to reduce the forcing and the change in <span class="inline-formula"><i>N</i>_i</span> due to anthropogenic emissions. The radiative forcing due to all increased anthropogenic emissions from PI to PD is estimated to be <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M17" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">0.20</mn><mo>±</mo><mn mathvariant="normal">0.05</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="64pt" height="10pt" class="svg-formula" dspmath="mathimg" md5hash="ee6782ae497c2753168f9e63fd644ae0"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00005.svg" width="64pt" height="10pt" src="acp-20-7801-2020-ie00005.png"/></svg:svg></span></span>&thinsp;W&thinsp;m<span class="inline-formula">⁻²</span>. If newly formed secondary organic aerosols (SOAs) act as INPs and inhibit homogeneous nucleation, the <span class="inline-formula"><i>N</i>_i</span> formed from heterogeneous nucleation is increased. As a result, the inclusion of INPs from SOA increases the change in <span class="inline-formula"><i>N</i>_i</span> to <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M21" display="inline" overflow="scroll" dspmath="mathml"><mrow><mn mathvariant="normal">12.0</mn><mo>±</mo><mn mathvariant="normal">2.3</mn><mo>×</mo><msup><mn mathvariant="normal">10</mn><mn mathvariant="normal">7</mn></msup></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="78pt" height="13pt" class="svg-formula" dspmath="mathimg" md5hash="e87119547d007917d733663ae1948ed7"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00006.svg" width="78pt" height="13pt" src="acp-20-7801-2020-ie00006.png"/></svg:svg></span></span>&thinsp;m<span class="inline-formula">⁻²</span> (9&thinsp;%) and increases (makes less negative) the anthropogenic forcing on cirrus clouds to <span class="inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" id="M23" display="inline" overflow="scroll" dspmath="mathml"><mrow><mo>-</mo><mn mathvariant="normal">0.04</mn><mo>±</mo><mn mathvariant="normal">0.08</mn></mrow></math><span><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="64pt" height="10pt" class="svg-formula" dspmath="mathimg" md5hash="c6d0e36cd9a44c954fcf10bf8403a4a2"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="acp-20-7801-2020-ie00007.svg" width="64pt" height="10pt" src="acp-20-7801-2020-ie00007.png"/></svg:svg></span></span>&thinsp;W&thinsp;m<span class="inline-formula">⁻²</span> from PI to PD.</p>